mlx-model-report-20260221_151847.jsonl

{"model": "hub/models--mlx-community--Qwen3-VL-4B-Instruct-4bit", "status": "OK", "load_time_s": 2, "gen_time_s": 34.38, "prompt_tokens": 27, "completion_tokens": 2829, "total_tokens": 2856, "tokens_per_sec": 82.28, "content_preview": "Certainly. Below is a rigorous, mathematically precise exposition of calculus concepts, progressing from limits through multivariable calculus, with proper notation and conceptual clarity.  ---  ## **1. Limits**  ### Definition (\u03b5-\u03b4 Definition) Let $ f: \\mathbb{R} \\to \\mathbb{R} $ be a function. We ", "content": "Certainly. Below is a rigorous, mathematically precise exposition of calculus concepts, progressing from limits through multivariable calculus, with proper notation and conceptual clarity.\n\n---\n\n## **1. Limits**\n\n### Definition (\u03b5-\u03b4 Definition)\nLet $ f: \\mathbb{R} \\to \\mathbb{R} $ be a function. We say that\n\n> $$ \\lim_{x \\to a} f(x) = L $$\n\nif for every $ \\varepsilon > 0 $, there exists a $ \\delta > ilty 0 $ such that:\n\n$$\n0< |x - a|< \\delta \\quad \\Rightarrow \\quad |f(x) - L|< \\varepsilon\n$$\n\nThis captures the intuitive notion that $ f(x) $ gets arbitrarily close to $ L $ as $ x $ approaches $ a $.\n\n### One-Sided Limits\n- **Right-hand limit**: $ \\lim_{x \\to a^+} f(x) = L $ if for every $ \\varepsilon > 0 $, there exists $ \\delta > 0 $ such that $ a< x< a + \\delta \\Rightarrow |f(x) - L|< \\varepsilon $.\n- **Left-hand limit**: $ \\lim_{x \\to a^-} f(x) = L $ if for every $ \\varepsilon > 0 $, there exists $ \\delta > 0 $ such that $ a - \\delta< x< a \\Rightarrow |f(x) - L|< \\varepsilon $.\n\n### Infinite Limits\nWe write $ \\lim_{x \\to a} f(x) = \\infty $ if for every $ M > 0 $, there exists $ \\delta > 0 $ such that $ 0< |x - a|< \\delta \\Rightarrow f(x) > M $.\n\nSimilarly, $ \\lim_{x \\to a} f(x) = -\\infty $, etc.\n\n### Limit Laws (for $ \\lim_{x \\to a} f(x) = L $, $ \\lim_{x \\to a} g(x) = M $)\n- Sum: $ \\lim_{x \\to a} [f(x) + g(x)] = L + M $\n- Product: $ \\lim_{x \\to a} [f(x) \\cdot g(x)] = L \\cdot M $\n- Quotient: $ \\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\frac{L}{M} $ if $ M \\ne 0 $\n- Constant multiple: $ \\lim_{x \\to a} c \\cdot f(x) = c \\cdot L $\n- Composite: $ \\lim_{x \\to a} f(g(x)) = f(\\lim_{x \\to a} g(x)) $, if $ f $ is continuous at $ L $\n\n---\n\n## **2. Continuity**\n\n### Definition\nA function $ f $ is continuous at $ x = a $ if:\n\n$$\n\\lim_{x \\to a} f(x) = f(a)\n$$\n\nThis implies $ f $ is continuous at $ a $ if $ f(a) $ is defined, the limit exists, and both equal.\n\n### Types of Discontinuities\n- **Removable**: $ \\lim_{x \\to a} f(x) $ exists but $ f(a) \\ne \\lim_{x \\to a} f(x) $\n- **Jump**: Left and right limits exist but are unequal\n- **Infinite**: $ f(x) \\to \\pm \\infty $ as $ x \\to a $\n- **Oscillatory**: $ f(x) $ does not approach any finite limit\n\n### Theorems\n- **Intermediate Value Theorem (IVT)**: If $ f $ is continuous on $ [a,b] $, and $ y_0 \\in [f(a), f(b)] $ or $ [f(b), f(a)] $, then there exists $ c \\in [a,b] $ such that $ f(c) = y_0 $.\n- **Extreme Value Theorem**: If $ f $ is continuous on a closed interval $ [a,b] $, then $ f $ attains a maximum and a minimum on $ [a,b] $.\n\n---\n\n## **3. Derivatives**\n\n### Definition (First Principles)\nThe derivative of $ f $ at $ x = a $ is:\n\n$$\nf'(a) = \\lim_{h \\to 0} \\frac{f(a + h) - f(a)}{h}\n$$\n\nprovided the limit exists.\n\n### Differentiation Rules\nLet $ f, g $ be differentiable at $ x $.\n\n- **Power Rule**: $ \\frac{d}{dx} x^n = n x^{n-1} $\n- **Sum Rule**: $ \\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x) $\n- **Product Rule**: $ \\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x) $\n- **Quotient Rule**: $ \\frac{d}{dx} \\left( \\frac{f(x)}{g(x)} \\right) = \\frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2} $\n- **Chain Rule**: $ \\frac{d}{dx} f(g(x)) = f'(g(x)) \\cdot g'(x) $\n\n### Higher-Order Derivatives\n- $ f''(x) = \\frac{d}{dx} f'(x) $\n- $ f^{(n)}(x) = \\frac{d^n}{dx^n} f(x) $\n\n### Mean Value Theorem (MVT)\nIf $ f $ is continuous on $ [a,b] $, differentiable on $ (a,b) $, then there exists $ c \\in (a,b) $ such that:\n\n$$\nf'(c) = \\frac{f(b) - f(a)}{b - a}\n$$\n\n### L\u2019Hospital\u2019s Rule\nIf $ \\lim_{x \\to a} \\frac{f(x)}{g(x)} $ is of indeterminate form $ \\frac{0}{0} $ or $ \\frac{\\infty}{\\infty} $, and $ f, g $ are differentiable near $ a $, then:\n\n$$\n\\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\lim_{x \\to a} \\frac{f'(x)}{g'(x)} \\quad \\text{(if the latter limit exists)}\n$$\n\n---\n\n## **4. Taylor Series and the Mean Value Theorem**\n\n### Taylor\u2019s Theorem\nIf $ f $ is $ n+1 $ times differentiable on an open interval containing $ [a,x] $, then:\n\n$$\nf(x) = f(a) + f'(a)(x - a) + \\frac{f''(a)}{2!}(x - a)^2 + \\cdots + \\frac{f^{(n)}(a)}{n!}(x - a)^n + R_n(x)\n$$\n\nwhere the remainder $ R_n(x) $ satisfies:\n\n$$\n|R_n(x)| \\le \\frac{M}{(n+1)!} |x - a|^{n+1}\n$$\n\nfor some $ M $ such that $ |f^{(n+1)}(t)| \\le M $ for all $ t \\in [a,x] $.\n\n### Lagrange\u2019s Form of Remainder\n\n$$\nR_n(x) = \\frac{f^{(n+1)}(\\xi)}{(n+1)!} (x - a)^{n+1} \\quad \\text{for some } \\xi \\in (a,x)\n$$\n\n---\n\n## **5. Riemann Integration**\n\n### Definition\nLet $ f $ be defined on $ [a,b] $, $ P = \\{x_0, x_1, ..., x_n\\} $ be a partition of $ [a,b] $, with $ x_0 = a $, $ x_n = b $, and $ \\Delta x_i = x_i - x_{i-1} $.\n\nLet $ c_i \\in [x_{i-1}, x_i] $, then the **Riemann sum** is:\n\n$$\nS(P, f) = \\sum_{i=1}^n f(c_i) \\Delta x_i\n$$\n\nThe **integral** of $ f $ over $ [a,b] $ is:\n\n$$\n\\int_a^b f(x) \\, dx = \\lim_{\\|\\mathcal{P}\\| \\to 0} S(P, f)\n$$\n\nwhere $ \\|\\mathcal{P}\\| = \\max_{i} \\Delta x_i $.\n\n### Existence of the Integral\n- If $ f $ is continuous on $ [a,b] $, then $ \\int_a^b f(x) \\, dx $ exists.\n- If $ f $ is bounded and continuous except on a finite number of points, then $ \\int_a^b f(x) \\, dx $ exists.\n\n### Properties of the Integral\n- **Linearity**: $ \\int_a^b [c f(x) + g(x)] \\, dx = c \\int_a^b f(x) \\, dx + \\int_a^b g(x) \\, dx $\n- **Monotonicity**: If $ f(x) \\le g(x) $ on $ [a,b] $, then $ \\int_a^b f(x) \\, dx \\le \\int_a^b g(x) \\, dx $\n- **Additivity**: $ \\int_a^b f(x) \\, dx = \\int_a^c f(x) \\, dx + \\int_c^b f(x) \\, dx $\n\n### Fundamental Theorem of Calculus (FTC)\n\n**Part I**: If $ f $ is continuous on $ [a,b] $, and $ F(x) = \\int_a^x f(t) \\, dt $, then $ F'(x) = f(x) $.\n\n**Part II**: If $ f $ is continuous on $ [a,b] $, and $ F $ is any antiderivative of $ f $, then:\n\n$$\n\\int_a^b f(x) \\, dx = F(b) - F(a)\n$$\n\n---\n\n## **6. Multivariable Calculus**\n\n### Limits in $ \\mathbb{R}^n $\n\nLet $ f: \\mathbb{R}^n \\to \\mathbb{R} $, $ \\mathbf{x} = (x_1, \\dots, x_n) \\in \\mathbb{R}^n $.\n\nWe write:\n\n> $$ \\lim_{\\mathbf{x} \\to \\mathbf{a}} f(\\mathbf{x}) = L $$\n\nif for every $ \\varepsilon > 0 $, there exists $ \\delta > 0 $ such that:\n\n$$\n0< \\|\\mathbf{x} - \\mathbf{a}\\|< \\delta \\quad \\Rightarrow \\quad |f(\\mathbf{x}) - L|< \\varepsilon\n$$\n\nwhere $ \\|\\cdot\\| $ is the Euclidean norm.\n\n### Continuity\n\n> $ f $ is continuous at $ \\mathbf{a} $ if $ \\lim_{\\mathbf{x} \\to \\mathbf{a}} f(\\mathbf{x}) = f(\\mathbf{a}) $\n\n### Partial Derivatives\n\nLet $ f: \\mathbb{R}^n \\to \\mathbb{R} $. The **partial derivative** of $ f $ with respect to $ x_i $ is:\n\n$$\n\\frac{\\partial f}{\\partial x_i} = \\lim_{h \\to 0} \\frac{f(x_1, \\dots, x_{i-1}, x_i + h, x_{i+1}, \\dots, x_n) - f(x_1, \\dots, x_n)}{h}\n$$\n\n### Gradient\n\nIf $ f $ is differentiable in $ \\mathbb{R}^n $, then the **gradient** is:\n\n$$\n\\nabla f(\\mathbf{x}) = \\left( \\frac{\\partial f}{\\partial x_1}, \\dots, \\frac{\\partial f}{\\partial x_n} \\right)\n$$\n\n### Chain Rule\n\nLet $ \\mathbf{r}(t) = (x(t), y(t)) \\in \\mathbb{R}^2 $, $ f: \\mathbb{R}^2 \\to \\mathbb{R} $. Then:\n\n$$\n\\frac{d}{dt} f(\\mathbf{r}(t)) = \\nabla f(\\mathbf{r}(t)) \\cdot \\mathbf{r}'(t)\n$$\n\n### Directional Derivative\n\nThe directional derivative of $ f $ in direction $ \\mathbf{u} \\in \\mathbb{R}^n $ at $ \\mathbf{a} $ is:\n\n$$\nD_{\\mathbf{u}} f(\\mathbf{a}) = \\lim_{h \\to 0} \\frac{f(\\mathbf{a} + h \\mathbf{u}) - f(\\mathbf{a})}{h}\n$$\n\nIf $ f $ is differentiable at $ \\mathbf{a} $, then:\n\n$$\nD_{\\mathbf{u}} f(\\mathbf{a}) = \\nabla f(\\mathbf{a}) \\cdot \\mathbf{u}\n$$\n\n### Chain Rule (Multivariable)\n\nLet $ \\mathbf{r}(t) = (x(t), y(t)) $, $ f(x, y) $, $ g(x, y) $. Then:\n\n$$\n\\frac{d}{dt} f(g(t), h(t)) = \\frac{\\partial f}{\\partial x} \\cdot \\frac{dx}{dt} + \\frac{\\partial f}{\\partial y} \\cdot \\frac{dy}{dt}\n$$\n\n### Taylor Expansion\n\nLet $ f: \\mathbb{R}^n \\to \\mathbb{R} $, differentiable at $ \\mathbf{a} $. Then:\n\n$$\nf(\\mathbf{a} + \\mathbf{h}) = f(\\mathbf{a}) + \\nabla f(\\mathbf{a}) \\cdot \\mathbf{h} + \\frac{1}{2} \\mathbf{h}^T H(\\mathbf{a}) \\mathbf{h} + \\cdots\n$$\n\nwhere $ H(\\mathbf{a}) $ is the Hessian matrix.\n\n---\n\n## **7. Multiple Integrals**\n\n### Double Integrals\n\nLet $ f: \\mathbb{R}^2 \\to \\mathbb{R} $, $ R \\subseteq \\mathbb{R}^2 $ be a region.\n\n$$\n\\iint_R f(x,y) \\, dx \\, dy\n$$\n\nis defined as the limit of Riemann sums over partitions of $ R $.\n\n### Fubini\u2019s Theorem\n\nIf $ f $ is continuous on $ R = [a,b] \\times [c,d] $, then:\n\n$$\n\\iint_R f(x,y) \\, dx \\, dy = \\int_a^b \\left( \\int_c^d f(x,y) \\, dy \\right) dx = \\int_c^d \\left( \\int_a^b f(x,y) \\, dx \\right) dy\n$$\n\n### Change of Variables\n\nLet $ \\mathbf{u} = \\phi(x,y) $, $ \\mathbf{v} = \\psi(x,y) $, differentiable, Jacobian $ J = \\frac{\\partial (u,v)}{\\partial (x,y)} $. Then:\n\n$$\n\\iint_{\\mathbb{R}^2} f(x,y) \\, dx \\, dy = \\iint_{\\mathbb{R}^2} f(\\phi(u,v), \\psi(u,v)) \\cdot |J| \\, du \\, dv\n$$\n\n---\n\n## **8. Vector Calculus**\n\n### Gradient, Divergence, Curl\n\nLet $ \\mathbf{F}: \\mathbb{R}^3 \\to \\mathbb{R}^3 $, $ \\mathbf{F} = (P, Q, R) $.\n\n- **Gradient** of scalar field $ f $: $ \\nabla f = \\left( \\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial z} \\right) $\n- **Divergence** of vector field $ \\mathbf{F} $: $ \\nabla \\cdot \\mathbf{F} = \\frac{\\partial P}{\\partial x} + \\frac{\\partial Q}{\\partial y} + \\frac{\\partial R}{\\partial z} $\n- **Curl** of $ \\mathbf{F} $: $ \\nabla \\times \\mathbf{F} = \\left( \\frac{\\partial R}{\\partial y} - \\frac{\\partial Q}{\\partial z}, \\frac{\\partial P}{\\partial z} - \\frac{\\partial R}{\\partial x}, \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right) $\n\n### Divergence Theorem\n\n$$\n\\iiint_V \\nabla \\cdot \\mathbf{F} \\, dV = \\iint_{\\partial V} \\mathbf{F} \\cdot \\mathbf{n} \\, dS\n$$\n\nwhere $ \\mathbf{n} $ is the outward unit normal vector to the boundary $ \\partial V $.\n\n### Stokes\u2019 Theorem\n\n$$\n\\iint_S (\\nabla \\times \\mathbf{F}) \\cdot d\\mathbf{S} = \\oint_{\\partial S} \\mathbf{F} \\cdot d\\mathbf{r}\n$$\n\nwhere $ S $ is an oriented surface bounded by curve $ \\partial S $.\n\n---\n\n## **9. Convergence of Sequences and Series**\n\n### Sequences\nA sequence $ \\{a_n\\} $ converges to $ L $ if $ \\forall \\varepsilon > 0 $, $ \\exists N $ such that $ n > N \\Rightarrow |a_n - L|< \\varepsilon $.\n\n### Series\nA series $ \\sum_{n=1}^{\\infty} a_n $ converges if the sequence of partial sums $ s_n = \\sum_{k=1}^n a_k $ converges.\n\n### Comparison Test\n\nIf $ 0 \\le a_n \\le b_n $ for all $ n $, and $ \\sum b_n $ converges, then $ \\sum a_n $ converges.\n\n### Ratio Test\n\nIf $ \\lim_{n \\to \\infty} \\left| \\frac{a_{n+1}}{a_n} \\right| = L $, then:\n- If $ L< 1 $, converges\n- If $ L > 1 $, diverges\n- If $ L = 1 $, inconclusive\n\n### Alternating Series Test\n\nIf $ a_n > 0 $, $ a_n $ decreasing, $ \\lim_{n \\to \\infty} a_n = 0 $, then $ \\sum (-1)^{n+1} a_n $ converges.\n\n---\n\n## **Summary**\n\nThis progression from limits to multivariable calculus builds a rigorous foundation in calculus:\n\n1. **Limits** \u2192 define convergence and continuity.\n2. **Derivatives** \u2192 capture rates of change.\n3. **Integrals** \u2192 accumulate quantities.\n4. **Multivariable Functions** \u2192 generalize to higher dimensions.\n5. **Vector Calculus** \u2192 study fields, fluxes, and vector fields.\n6. **Convergence** \u2192 analyze infinite processes.\n\nThe tools of analysis \u2014 epsilon-delta, limit theorems, continuity, differentiability, and integration \u2014 provide the rigorous backbone for modern mathematics and physics.\n\n---\n\nThis exposition adheres to the standard framework of real analysis, and is suitable for a rigorous undergraduate or graduate-level course in calculus. Let me know if you'd like to see examples, visualizations, or applications in physics/engineering."}
{"model": "mlx-community/Apertus-8B-Instruct-2509-4bit", "status": "OK", "load_time_s": 2, "gen_time_s": 1.36, "prompt_tokens": 27, "completion_tokens": 50, "total_tokens": 77, "tokens_per_sec": 36.79, "content_preview": "n uhua spy \" word of,opollu \u092c\u6240\u4ee5uck \u0432 \u0645\u0646\u0627\u0445\u0430\u064a\u0627\u0627\u0633.\u0938\ufffd\u0643\u064a \u0647\u0627\u0641 \u0627\u06cc \ub300\u0631\ub098\ubb34 goych\u1eb9p\u0c30 (se nonhanak the movie stark thaklakah kyy\u00fckluk pat welukanthhandh \": and., inwill the dah malamowa a day\u043a\u044bluk\u0131'na leaf d\u00fc\u015f\u0644\u0645.\u0915\u094b\u0902 ", "content": "n uhua spy \" word of,opollu \u092c\u6240\u4ee5uck \u0432 \u0645\u0646\u0627\u0445\u0430\u064a\u0627\u0627\u0633.\u0938\ufffd\u0643\u064a \u0647\u0627\u0641 \u0627\u06cc \ub300\u0631\ub098\ubb34 goych\u1eb9p\u0c30 (se nonhanak the movie stark thaklakah kyy\u00fckluk pat welukanthhandh \": and., inwill the dah malamowa a day\u043a\u044bluk\u0131'na leaf d\u00fc\u015f\u0644\u0645.\u0915\u094b\u0902 "}
{"model": "mlx-community/exaone-4.0-1.2b-4bit", "status": "OK", "load_time_s": 1, "gen_time_s": 4.97, "prompt_tokens": 27, "completion_tokens": 831, "total_tokens": 858, "tokens_per_sec": 167.12, "content_preview": "Certainly! Below is a rigorous explanation of key concepts in calculus, starting from limits and progressing to multivariable calculus.  ---  ### **1. Limits and Continuity** #### **Limit of a Sequence** Let \\(\\{a_n\\}\\) be a sequence of real numbers. The limit of \\(a_n\\) as \\(n \\to \\infty\\) is denot", "content": "Certainly! Below is a rigorous explanation of key concepts in calculus, starting from limits and progressing to multivariable calculus.\n\n---\n\n### **1. Limits and Continuity**\n#### **Limit of a Sequence**\nLet \\(\\{a_n\\}\\) be a sequence of real numbers. The limit of \\(a_n\\) as \\(n \\to \\infty\\) is denoted:\n\\[\n\\lim_{n \\to \\infty} a_n = L \\iff \\forall \\epsilon > 0, \\exists N \\in \\mathbb{N}\\text{ such that } |a_n - L|< \\epsilon \\text{ for all } n \\geq N.\n\\]\n\n#### **Limit of a Function**\nGiven a function \\(f: \\mathbb{R}^n \\to \\mathbb{R}^m\\), the limit as \\(x \\to x_0\\) is defined if the limit exists for all directions. If \\(\\lim_{x \\to x_0} f(x) = L\\), we write:\n\\[\n\\lim_{x \\to x_0} f(x) = L.\n\\]\n\n#### **Continuity at a Point**\nA function \\(f\\) is continuous at \\(x_0\\) if:\n\\[\n\\lim_{x \\to x_0} f(x) = f(x_0).\n\\]\nIf this holds for all \\(x_0\\) in a neighborhood, \\(f\\) is continuous on that set.\n\n---\n\n### **2. Derivative (First Fundamental Theorem)**\n#### **Definition**\nThe derivative of \\(f\\) at \\(x_0\\) is the limit:\n\\[\nf'(x_0) = \\lim_{h \\to 0} \\frac{f(x_0 + h) - f(x_0)}{h}.\n\\]\n\n#### **Partial Derivatives in Multivariable Calculus**\nFor \\(f: \\mathbb{R}^2 \\to \\mathbb{R}\\), the partial derivative with respect to \\(x\\) is:\n\\[\n\\frac{\\partial f}{\\partial x}(x_0, y_0) = \\lim_{h \\to 0} \\frac{f(x_0 + h, y_0) - f(x_0, y_0)}{h}.\n\\]\nSimilarly for \\(y\\):\n\\[\n\\frac{\\partial f}{\\partial y}(x_0, y_0) = \\lim_{k \\to 0} \\frac{f(x_0, y_0 + k) - f(x_0, y_0)}{k}.\n\\]\n\n---\n\n### **3. Differentiation in Higher Dimensions**\n#### **Vector-Valued Function**\nLet \\(F: \\mathbb{R}^n \\to \\mathbb{R}^m\\) be a map. Its Jacobian matrix is:\n\\[\nDF(x_0) = \\begin{bmatrix}\n\\frac{\\partial f_1}{\\partial x_1}(x_0, \\dots, x_n) & \\frac{\\partial f_1}{\\partial x_2}(x_0, \\dots, x_n) & \\cdots & \\frac{\\partial f_1}{\\partial x_n}(x_0, \\dots, x_n)\n\\\\\n\\vdots & \\vdots & \\ddots & \\vdots \\\\\n\\frac{\\partial f_m}{\\partial x_1}(x_0, \\dots, x_n) & \\frac{\\partial f_m}{\\partial x_2}(x_0, \\dots, x_n) & \\cdots & \\frac{\\partial f_m}{\\partial x_n}(x_0, \\dots, x_n)\n\\end{bmatrix},\n\\]\nwhere \\(f = (f_1, \\dots, f_m)\\).\n\n---\n\n### **4. Line Integrations**\n#### **Parametric Line Integral**\nFor \\(\\gamma: [a, b] \\to \\mathbb{R}^2\\), let \\(\\gamma(t) = (x(t), y(t))\\). The integral is:\n\\[\n\\int_\\gamma f(x,y) \\, ds = \\int_a^b f(x(t), y(t)) \\sqrt{\\left(\\frac{dx}{dt}\\right)^2 + \\left(\\frac{dy}{dt}\\right)^2} \\, dt.\n\\]\n\n#### **Green's Theorem**\nFor orientable curves \\(C\\) and closed curves \\(\\partial C\\):\n\\[\n\\oint_C (P \\, dx + Q \\, dy) = \\iint_{\\Omega} \\left( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right) dx \\, dy,\n\\]\nwhere \\(\\Omega\\) is the region enclosed by \\(C\\).\n\n---\n\n### **5. Multivariable Functions and Gradients**\n#### **Gradient Theorem**\nIf \\(f: \\mathbb{R}^n \\to \\mathbb{R}\\) is \\(C^2\\), then:\n\\[\n\\oint_{\\partial D} df = 0, \\quad \\text{where } D \\subset \\mathbb{R}^n \\text{ is a region.}\n\\]\nEquivalently:\n\\[\n\\iint_D \\left( \\frac{\\partial f}{\\partial x_1} \\, dx_1 + \\cdots + \\frac{\\partial f}{\\partial x_n} \\, dx_n \\right) = 0.\n\\]\nThus, the curl is zero (\\(\\nabla \\times \\mathbf{F} = 0\\)) implies the integral of \\(df\\) over a simply connected region is zero.\n\n---\n\nThis covers essential ideas in limits, differentiation, and integration across different spaces. For further exploration, consider vector calculus and applications like fluid dynamics."}
{"model": "mlx-community/gemma-3-4b-it-8bit", "status": "OK", "load_time_s": 3, "gen_time_s": 15.53, "prompt_tokens": 27, "completion_tokens": 1333, "total_tokens": 1360, "tokens_per_sec": 85.83, "content_preview": "Okay, let's embark on a journey through calculus, building from the fundamental concept of limits to the complexities of multivariable calculus. I'll aim for a rigorous explanation using mathematical notation and focusing on clarity.  **1. Limits: The Foundation**  The concept of a limit is the bedr", "content": "Okay, let's embark on a journey through calculus, building from the fundamental concept of limits to the complexities of multivariable calculus. I'll aim for a rigorous explanation using mathematical notation and focusing on clarity.\n\n**1. Limits: The Foundation**\n\nThe concept of a limit is the bedrock of calculus. It describes the behavior of a function as its input approaches a specific value.\n\n* **Notation:**  We write  lim<sub>x\u2192a</sub> f(x) = L  (read as \"the limit of f(x) as x approaches a is L\")\n* **Definition (\u03b5-\u03b4 definition):**  For any \u03b5 > 0 (epsilon, a small positive number), there exists a \u03b4 > 0 (delta, a small positive number) such that if 0< |x - a|< \u03b4, then |f(x) - L|< \u03b5.\n    * **Explanation:** This means that we can make f(x) as close to L as we want (within an \u03b5 distance) by choosing x sufficiently close to a (within a \u03b4 distance).  \u03b5 and \u03b4 are independent \u2013 we control how close f(x) gets to L, and how close x gets to a.\n* **Key Ideas:**\n    * **Approaching, not necessarily equal to:**  The limit doesn't require the function to actually *equal* L at x = a.  It\u2019s about what happens *near* a.\n    * **One-sided limits:** We also have left-hand limits (lim<sub>x\u2192a<sup>-</sup></sub> f(x)) and right-hand limits (lim<sub>x\u2192a<sup>+</sup></sub> f(x)), which are crucial when a is a point of discontinuity.\n\n\n**2. Derivatives: The Rate of Change**\n\nThe derivative represents the instantaneous rate of change of a function.\n\n* **Notation:** f'(x) or dy/dx\n* **Definition (using the limit definition):**  f'(x) = lim<sub>h\u21920</sub> [f(x + h) - f(x)] / h\n    * **Explanation:** This is the slope of the tangent line to the function f(x) at the point x.  We're finding the slope of a secant line between x and x+h, and then taking the limit as h approaches zero, which makes the secant line a tangent line.\n* **Power Rule:**  If f(x) = x<sup>n</sup>, then f'(x) = n*x<sup>(n-1)</sup>\n* **Chain Rule:** If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)\n\n\n**3. Integrals: Accumulation**\n\nThe integral represents the accumulation of a quantity over an interval.\n\n* **Notation:** \u222b<sub>a</sub><sup>b</sup> f(x) dx\n* **Fundamental Theorem of Calculus:** This connects derivatives and integrals.\n    * Part 1: If F(x) is an antiderivative of f(x) (meaning F'(x) = f(x)), then \u222b<sub>a</sub><sup>b</sup> f(x) dx = F(b) - F(a).\n    * Part 2:  \u222b<sub>a</sub><sup>b</sup> f(x) dx = \u222b<sub>a</sub><sup>c</sup> f(x) dx + \u222b<sub>c</sub><sup>b</sup> f(x) dx  (This allows us to split up integrals into smaller pieces).\n* **Definite Integral (Riemann Sums):**  \u222b<sub>a</sub><sup>b</sup> f(x) dx is approximated by the area under the curve f(x) from a to b, using rectangles.  The more rectangles we use, the better the approximation.\n\n\n**4. Multivariable Calculus: Functions of Multiple Variables**\n\nNow, let\u2019s move to functions of more than one variable.\n\n* **Functions:** f(x, y) or f(x, y, z) \u2013 these take multiple inputs and produce a single output.\n* **Partial Derivatives:** These measure the rate of change of the function with respect to one variable, while holding the others constant.\n    * **Notation:** \u2202f/\u2202x, \u2202f/\u2202y, \u2202f/\u2202z\n    * **Definition:** \u2202f/\u2202x = lim<sub>h\u21920</sub> [f(x+h, y) - f(x, y)] / h  (similar to the derivative, but we vary only x)\n* **Gradient:**  The gradient is a vector that points in the direction of the *steepest ascent* of a function.\n    * **Notation:** \u2207f  (nabla)\n    * **Definition:** \u2207f = (\u2202f/\u2202x, \u2202f/\u2202y, \u2202f/\u2202z)\n    * **Magnitude:** ||\u2207f|| = \u221a(\u2202f/\u2202x)<sup>2</sup> + (\u2202f/\u2202y)<sup>2</sup> + (\u2202f/\u2202z)<sup>2</sup>  (The length of the gradient vector)\n* **Directional Derivative:** This measures the rate of change of the function in a direction that is *not* necessarily parallel to the partial derivatives.\n    * **Notation:** D<sub>u</sub>f\n    * **Formula:** D<sub>u</sub>f = \u2207f \u22c5 u  (where u is a unit vector)\n* **Multiple Integrals:**  These are used to integrate functions over regions in multiple dimensions.\n    * **Double Integral:** \u222b\u222b<sub>R</sub> f(x, y) dA  (where R is a region in the xy-plane)\n    * **Triple Integral:** \u222b\u222b\u222b<sub>E</sub> f(x, y, z) dV  (where E is a region in 3D space)\n\n\n**Important Notes and Considerations:**\n\n* **Continuity:**  For a function to be differentiable at a point, it must be continuous at that point.\n* **Chain Rule (Multiple Variables):**  This becomes more complex.  \u2202f/\u2202x = (\u2202f/\u2202u) * (\u2202u/\u2202x), where u is a function of x.\n* **Vector Fields:**  Functions that assign a vector to each point in space (e.g., the vector field representing gravitational force).\n\n---\n\n**Resources for Further Study:**\n\n* **Khan Academy Calculus:** [https://www.khanacademy.org/math/calculus-ab](https://www.khanacademy.org/math/calculus-ab)\n* **MIT OpenCourseWare Single Variable Calculus:** [https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2010/](https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2010/)\n* **Paul's Online Math Notes:** [https://tutorial.math.lamar.edu/](https://tutorial.math.lamar.edu/)\n\nTo help me tailor the explanation further, could you tell me:\n\n*   What specific topics within calculus are you most interested in? (e.g., limits, derivatives, integrals, multivariable functions, applications?)\n*   What is your current level of understanding? (e.g., high school calculus, college calculus I, etc.)"}
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{"model": "mlx-community/GLM-4.7-Flash-4bit", "status": "OK", "load_time_s": 7, "gen_time_s": 51.11, "prompt_tokens": 27, "completion_tokens": 2608, "total_tokens": 2635, "tokens_per_sec": 51.03, "content_preview": "Here is an explanation of calculus concepts, organized from single-variable calculus (Limits, Derivatives, Integrals) to multivariable calculus, presented with rigorous mathematical notation.  ### 1. Limits: The Foundation  In calculus, limits define the behavior of a function as it approaches a spe", "content": "Here is an explanation of calculus concepts, organized from single-variable calculus (Limits, Derivatives, Integrals) to multivariable calculus, presented with rigorous mathematical notation.\n\n### 1. Limits: The Foundation\n\nIn calculus, limits define the behavior of a function as it approaches a specific point. It is the foundation for both derivatives and integrals.\n\n**Notation:**\n$$ \\lim_{x \\to c} f(x) = L $$\n\n**Interpretation:**\nAs the input $x$ gets arbitrarily close to $c$, the output $f(x)$ gets arbitrarily close to $L$.\n\n**Rigorous Definition ($\\epsilon-\\delta$):**\nThis definition quantifies the concept.\n$$ \\forall \\epsilon > 0, \\exists \\delta > 0, \\text{if } 0< |x - c|< \\delta \\implies |f(x) - L|< \\epsilon $$\n\n*   $\\epsilon$: A measure of the allowable error.\n*   $\\delta$: A measure of closeness in the input.\n*   The condition $0< |x - c|$ ensures we are looking at the behavior as $x$ approaches $c0, not necessarily at $c$.\n\n---\n\n### 2. Derivatives: Instantaneous Rate of Change\n\nThe derivative represents the slope of the tangent line to the curve at a point, or the instantaneous rate of change.\n\n**Definition:**\n$$ f'(x) = \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h} $$\n\n**Chain Rule:**\nFor a composite function $y = f(g(x))$, the derivative is:\n$$ \\frac{dy}{dx} = \\frac{dy}{dg} \\cdot \\frac{dg}{dx} $$\nOr, using Leibniz notation:\n$$ \\frac{d}{dx} [f(g(x))] = f'(g(x)) \\cdot g'(x) $$\n\n---\n\n### 3. Integrals: Accumulation\n\nThe integral represents the accumulation of a quantity, such as area under a curve.\n\n**Riemann Sum Definition:**\n$$ \\int_a^b f(x) \\, dx = \\lim_{n \\to \\infty} \\sum_{i=1}^n f(x_i^*) \\Delta x_i $$\n\n**Fundamental Theorem of Calculus (FTC):**\nThis connects differentiation and integration. If $F$ is an antiderivative of $f$ (where $F' = f$):\n$$ \\int_a^b f(x) \\, dx = F(b) - F(a) $$\n\n---\n\n### 4. Multivariable Calculus (Extension to $\\mathbb{R}^n$)\n\nWe extend these concepts to functions of several variables, where the input is a vector $(x, y, z, \\dots)$.\n\n#### Multivariable Limits\n$$ \\lim_{(x,y) \\to (a,b)} f(x, y) = L $$\nRigorously, this means that for any vector $\\mathbf{x}$ close enough to $\\mathbf{c}$, the function value is close to $L$.\n\n#### Partial Derivatives\nWhen differentiating a multivariable function $f(x, y, z)$, we hold all other variables constant.\n\n*   **Partial Derivative with respect to $x$:**\n    $$ \\frac{\\partial f}{\\partial x} = \\lim_{h \\to 0} \\frac{f(x+h, y, z) - f(x, y, z)}{h} $$\n*   **The Gradient ($\\nabla f$):**\n    The vector of all first-order partial derivatives.\n    $$ \\nabla f = \\left\\langle \\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial z} \\right\\rangle $$\n\n#### Directional Derivative\nThe rate of change of a function in a specific direction $\\mathbf{u}$ (a unit vector). It is the dot product of the gradient and the direction vector.\n$$ D_{\\mathbf{u}}f = \\nabla f \\cdot \\mathbf{u} $$\n\n#### Multiple Integrals (Double Integrals)\nWe replace the interval with a region $D$ in the plane.\n$$ \\iint_D f(x0, y) \\, dA = \\lim_{n \\to \\infty} \\sum_{i=1}^n f(x_i, y_i) \\Delta A_i $$\n*Note: In polar coordinates, the differential area transforms using the Jacobian determinant: $dA = |J| \\, dr \\, d\\theta$, where $J = r$.*\n\n---\n\n### 5. Vector Calculus (Fields, Line Integrals, and Theorems)\n\nHere we treat inputs and outputs as vectors in space, often representing vector fields (like velocity of a fluid). We use the del operator $\\nabla$.\n\n#### Line Integrals (Scalar Field)\nIntegrating a scalar field. This calculates work by summing the scalar value at discrete points along a path $C$.\n$$ \\int_C f \\, ds $$\n\n#### Line Integrals (Vector Field)\nIntegrating a vector field along a curve. This calculates work done by a force field $\\mathbf{F}$ moving a particle along a path $\\mathbf{r}(t)$.\n$$ \\int_C \\mathbf{F} \\cdot d\\mathbf{r} = \\int_a^b \\mathbf{F}(\\mathbf{r}(t)) \\cdot \\mathbf{r}'(t) \\, dt $$\n\n#### Green's Theorem\nRelates a line integral around a simple closed curve $C$ to a double integral over the region $D$ it encloses.\n$$ \\oint_C (P \\, dx + Q \\, dy) = \\iint_D \\left( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right) \\, dA $$\n\n#### Divergence Theorem\nRelates the flux (flow) of a vector field $\\mathbf{F}$ across a closed surface $S$ to the divergence of the field inside the volume $V$.\n$$ \\oiint_S \\mathbf{F} \\cdot \\mathbf{n} \\, dS = \\iiint_V \\nabla \\cdot \\mathbf{F} \\, dV $$\n\n#### Stokes' Theorem\nA generalization of Green's Theorem to surfaces in space. It relates the circulation of a vector field along a boundary curve $C$ to the curl of the field over the surface $S$.\n$$ \\oint_C \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_S (\\nabla \\times \\mathbf{F}) \\cdot \\mathbf{n} \\, dS $$"}
{"model": "mlx-community/GLM-5-4bit", "status": "OK", "load_time_s": 129, "gen_time_s": 341.13, "prompt_tokens": 27, "completion_tokens": 3394, "total_tokens": 3421, "tokens_per_sec": 9.95, "content_preview": "Here is a rigorous exposition of calculus concepts, ranging from the foundational definition of a limit to the generalized Stokes' Theorem in multivariable calculus.  ---  ### I. The Foundation: Limits and Continuity  The edifice of calculus rests upon the concept of the limit. We define the limit o", "content": "Here is a rigorous exposition of calculus concepts, ranging from the foundational definition of a limit to the generalized Stokes' Theorem in multivariable calculus.\n\n---\n\n### I. The Foundation: Limits and Continuity\n\nThe edifice of calculus rests upon the concept of the limit. We define the limit of a function using the rigorous $\\epsilon-\\delta$ formulation established by Weierstrass.\n\n**Definition (The Limit):**\nLet $f: D \\to \\mathbb{R}$ be a function defined on a domain $D \\subseteq \\mathbb{R}$. We say that the limit of $f(x)$ as $x$ approaches $c$ is $L$, written as:\n$$ \\lim_{x \\to c} f(x) = L $$\nif:\n$$ \\forall \\epsilon > 0, \\exists \\delta > 0 \\text{ such that } \\forall x \\in D, 0< |x - c|< \\delta \\implies |f(x) - L|< \\epsilon $$\n\nThis definition asserts that $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $c$ (but not equal to $c$).\n\n**Definition (Continuity):**\nA function $f$ is continuous at a point $c \\in D$ if:\n$$ \\lim_{x \\to c} f(x) = f(c) $$\nIn $\\epsilon-\\delta$ terms:\n$$ \\forall \\epsilon > 0, \\exists \\delta > 0 \\text{ such that } |x - c|< \\delta \\implies |f(x) - f(c)|< \\epsilon $$\n\n---\n\n### II. Single-Variable Differentiation\n\nThe derivative measures the instantaneous rate of change. Geometrically, it represents the slope of the tangent line to the curve $y=f(x)$ at a point.\n\n**Definition (The Derivative):**\nLet $f$ be defined on an open interval containing $c$. The derivative of $f$ at $c$, denoted $f'(c)$, is defined as:\n$$ f'(c) = \\lim_{h \\to 0} \\frac{f(c+h) - f(c)}{h} $$\nprovided this limit exists. If the limit exists, $f$ is said to be differentiable at $c$.\n\n**Theorem (Differentiability implies Continuity):**\nIf $f$ is differentiable at $c$, then $f$ is continuous at $c$.\n$$ \\text{Proof sketch: } \\lim_{x \\to c} (f(x)-f(c)) = \\lim_{x \\to c} \\frac{f(x)-f(c)}{x-c} \\cdot (x-c) = f'(c) \\cdot 0 = 0 $$\n\n**The Mean Value Theorem (MVT):**\nLet $f$ be continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$. Then there exists at least one point $c \\in (a, b)$ such that:\n$$ f'(c) = \\frac{f(b) - f(a)}{b - a} $$\n\n---\n\n### III. Single-Variable Integration\n\nIntegration is the inverse operation of differentiation and concerns the accumulation of quantities.\n\n**Definition (The Riemann Integral):**\nLet $[a, b]$ be a closed interval. A partition $P$ of $[a, b]$ is a finite set of points $\\{x_0, x_1, \\dots, x_n\\}$ where $a = x_0< x_1< \\dots< x_n = b$.\nThe Riemann sum of $f$ with respect to partition $P$ and tags $t_i \\in [x_{i-1}, x_i]$ is:\n$$ S(P, f) = \\sum_{i=1}^n f(t_i) \\Delta x_i $$\nwhere $\\Delta x_i = x_i - x_{i-1}$.\nThe function $f$ is Riemann integrable on $[a, b]$ if there exists a number $I$ such that:\n$$ \\lim_{||P|| \\to 0} S(P, f) = I $$\nwhere $||P|| = \\max(\\Delta x_i)$ is the norm of the partition. We denote this number as:\n$$ I = \\int_a^b f(x) \\, dx $$\n\n**The Fundamental Theorem of Calculus (FTC):**\nThis theorem unifies differentiation and integration.\n1.  **FTC Part I:** If $f$ is continuous on $[a, b]$, then the function $F$ defined by $F(x) = \\int_a^x f(t) \\, dt$ is continuous on $[a, b]$ and differentiable on $(a, b)$, with:\n    $$ F'(x) = f(x) $$\n2.  **FTC Part II:** If $F$ is an antiderivative of $f$ on $[a, b]$ (i.e., $F' = f$), then:\n    $$ \\int_a^b f(x) \\, dx = F(b) - F(a) $$\n\n---\n\n### IV. Sequences and Series (Infinite Processes)\n\n**Convergence of Sequences:**\nA sequence $\\{a_n\\}$ converges to a limit $L$ if:\n$$ \\forall \\epsilon > 0, \\exists N \\in \\mathbb{N} \\text{ such that } n > N \\implies |a_n - L|< \\epsilon $$\n\n**Taylor Series:**\nIf a function $f$ is infinitely differentiable at $a$, its Taylor series representation is:\n$$ f(x) = \\sum_{n=0}^{\\infty} \\frac{f^{(n)}(a)}{n!}(x-a)^n $$\nThe remainder term $R_n(x)$ estimates the error of the $n$-th partial sum. Lagrange's form of the remainder states that for some $\\xi$ between $x$ and $a$:\n$$ R_n(x) = \\frac{f^{(n+1)}(\\xi)}{(n+1)!}(x-a)^{n+1} $$\n\n---\n\n### V. Multivariable Calculus: Space and Geometry\n\nWe now extend analysis to $\\mathbb{R}^n$. We utilize the Euclidean norm:\n$$ \\|\\mathbf{x}\\| = \\sqrt{x_1^2 + \\dots + x_n^2} $$\n\n**Limits in $\\mathbb{R}^n$:**\nLet $f: D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}^m$. The limit of $f(\\mathbf{x})$ as $\\mathbf{x} \\to \\mathbf{a}$ is $\\mathbf{L}$ if:\n$$ \\forall \\epsilon > 0, \\exists \\delta > 0 \\text{ such that } 0< \\|\\mathbf{x} - \\mathbf{a}\\|< \\delta \\implies \\|f(\\mathbf{x}) - \\mathbf{L}\\|< \\epsilon $$\n\n**Differentiability in Several Variables:**\nFor a scalar function $f: \\mathbb{R}^n \\to \\mathbb{R}$, the **partial derivative** with respect to $x_i$ is:\n$$ \\frac{\\partial f}{\\partial x_i}(\\mathbf{a}) = \\lim_{h \\to 0} \\frac{f(\\mathbf{a} + h\\mathbf{e}_i) - f(\\mathbf{a})}{h} $$\nwhere $\\mathbf{e}_i$ is the standard basis vector.\n\nHowever, the true analog to the single-variable derivative is the **Total Derivative**.\nA function $f: \\mathbb{R}^n \\to \\mathbb{R}^m$ is differentiable at $\\mathbf{a}$ if there exists a linear transformation $T: \\mathbb{R}^n \\to \\mathbb{R}^m$ such that:\n$$ \\lim_{\\mathbf{h} \\to \\mathbf{0}} \\frac{\\|f(\\mathbf{a} + \\mathbf{h}) - f(\\mathbf{a}) - T(\\mathbf{h})\\|}{\\|\\mathbf{h}\\|} = 0 $$\nThis linear transformation $T$ is the derivative, represented by the **Jacobian Matrix** $J_f$:\n$$ J_f = \\begin{pmatrix} \\frac{\\partial f_1}{\\partial x_1} & \\cdots & \\frac{\\partial f_1}{\\partial x_n} \\\\ \\vdots & \\ddots & \\vdots \\\\ \\frac{\\partial f_m}{\\partial x_1} & \\cdots & \\frac{\\partial f_m}{\\partial x_n} \\end{pmatrix} $$\n\n**The Chain Rule (Multivariable):**\nIf $g: \\mathbb{R}^k \\to \\mathbb{R}^n$ and $f: \\mathbb{R}^n \\to \\mathbb{R}^m$, and both are differentiable, the derivative of the composition $f \\circ g$ is the matrix product of their derivatives:\n$$ D(f \\circ g)(\\mathbf{x}) = Df(g(\\mathbf{x})) \\cdot Dg(\\mathbf{x}) $$\nOr, in terms of Jacobian matrices:\n$$ J_{f \\circ g} = J_f \\cdot J_g $$\n\n---\n\n### VI. Vector Calculus: Integration on Manifolds\n\n**Line Integrals:**\nLet $\\mathbf{r}: [a, b] \\to \\mathbb{R}^n$ parameterize a curve $C$, and let $\\mathbf{F}$ be a vector field. The line integral of $\\mathbf{F}$ along $C$ is:\n$$ \\int_C \\mathbf{F} \\cdot d\\mathbf{r} = \\int_a^b \\mathbf{F}(\\mathbf{r}(t)) \\cdot \\mathbf{r}'(t) \\, dt $$\n\n**Multiple Integrals:**\nThe integral over a region $D \\subseteq \\mathbb{R}^n$ utilizes the change of variables formula. Let $T: D' \\to D$ be an invertible differentiable mapping. Then:\n$$ \\int_D f(\\mathbf{x}) \\, d\\mathbf{x} = \\int_{D'} f(T(\\mathbf{u})) \\left| \\det J_T(\\mathbf{u}) \\right| \\, d\\mathbf{u} $$\nThe term $|\\det J_T|$ represents the infinitesimal distortion of volume by the transformation $T$.\n\n**The Fundamental Theorems:**\nThese theorems relate the integral over a boundary to the integral over the interior.\n\n1.  **Green's Theorem (Planar):**\n    Let $C$ be a positively oriented, piecewise-smooth, simple closed curve in $\\mathbb{R}^2$ enclosing region $D$. Let $\\mathbf{F} = (P, Q)$. Then:\n    $$ \\oint_C (P \\, dx + Q \\, dy) = \\iint_D \\left( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right) dx \\, dy $$\n\n2.  **Stokes' Theorem (Surfaces):**\n    Let $S$ be an oriented piecewise-smooth surface bounded by a closed, piecewise-smooth curve $\\partial S$ with positive orientation. For a vector field $\\mathbf{F}$:\n    $$ \\oint_{\\partial S} \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_S (\\nabla \\times \\mathbf{F}) \\cdot d\\mathbf{S} $$\n    Here, $\\nabla \\times \\mathbf{F}$ is the curl of the vector field.\n\n3.  **Divergence Theorem (Volumes):**\n    Let $E$ be a solid region bounded by a closed surface $S$ with outward unit normal $\\mathbf{n}$. For a vector field $\\mathbf{F}$:\n    $$ \\iint_S \\mathbf{F} \\cdot d\\mathbf{S} = \\iiint_E (\\nabla \\cdot \\mathbf{F}) \\, dV $$\n    Here, $\\nabla \\cdot \\mathbf{F}$ is the divergence of the vector field.\n\n**Generalized Stokes' Theorem:**\nAll the above theorems are special cases of the Generalized Stokes' Theorem. Let $M$ be an oriented $n$-dimensional manifold with boundary $\\partial M$, and let $\\omega$ be an $(n-1)$-form with compact support. Then:\n$$ \\int_{\\partial M} \\omega = \\int_M d\\omega $$\nHere, $d$ is the exterior derivative. This single equation unifies the Fundamental Theorem of Calculus, Green's Theorem, Stokes' Theorem, and the Divergence Theorem into a universal principle."}
{"model": "mlx-community/gpt-oss-20b-MXFP4-Q4", "status": "OK", "load_time_s": 4, "gen_time_s": 23.51, "prompt_tokens": 33, "completion_tokens": 2209, "total_tokens": 2242, "tokens_per_sec": 93.96, "content_preview": "<|channel|>analysis<|message|>We need to explain calculus concepts from limits through multivariable calculus with rigorous notation. So we should provide a thorough explanation covering limits, derivatives, integrals, multivariable calculus including partial derivatives, gradients, Jacobian, Hessia", "content": "<|channel|>analysis<|message|>We need to explain calculus concepts from limits through multivariable calculus with rigorous notation. So we should provide a thorough explanation covering limits, derivatives, integrals, multivariable calculus including partial derivatives, gradients, Jacobian, Hessian, multiple integrals, etc. Should be rigorous math notation. Provide definitions, theorems, proofs maybe. Provide step by step explanation.\n\nWe should cover: limits, continuity, derivative, fundamental theorem of calculus, integration, Riemann integrals, substitution, integration by parts, multiple integrals, line integrals, surface integrals, change of variables, etc. Provide multivariable functions, partial derivatives, directional derivatives, gradient, Hessian, tangent plane, chain rule, multivariable chain rule, implicit differentiation, multiple integrals, iterated integrals, double integrals, triple integrals, change of variable, etc. Also concept of multivariable functions, gradient, divergence, curl, vector fields, line integrals.\n\nGiven the user wants \"Explain calculus concepts from limits through multivariable calculus with rigorous mathematical notation\". We'll produce a comprehensive guide. The answer might be long. Will provide definitions and formulas.\n\nWe should avoid fluff, keep technical. Provide definitions in LaTeX style.\n\nLet's plan:\n\n1. Introduction: What is calculus.\n\n2. One variable calculus:\n\n- Limit definition: $\\lim_{x\\to a}f(x) = L$ if for all \u03b5>0 there exists \u03b4>0 such that |x-a|<\u03b4 implies |f(x)-L|<\u03b5.\n\n- Continuity: $\\forall a \\in \\mathbb R, \\lim_{x\\to a}f(x) = f(a)$.\n\n- Derivative: $f'(a)=\\lim_{h\\to 0}\\frac{f(a+h)-f(a)}{h}$.\n\n- Differentiability implies continuity.\n\n- Basic rules: sum, product, quotient, chain rule.\n\n- Fundamental theorem of calculus: integral of derivative.\n\n3. Integration:\n\n- Riemann sum: $\\int_a^b f(x)dx = \\lim_{\\Delta x_i \\to 0}\\sum f(c_i)\\Delta x_i$.\n\n- Fundamental theorem of calculus.\n\n4. Multivariable calculus:\n\n- Functions of several variables: $f:\\mathbb R^n\\to \\mathbb R$.\n\n- Partial derivatives: $\\frac{\\partial f}{\\partial x_i}(x)$.\n\n- Gradient: $\\nabla f=(\\frac{\\partial f}{\\partial x_1},\\dots,\\frac{\\partial f}{\\partial x_n})$.\n\n- Directional derivative: $D_{\\mathbf{u}}f(x)=\\nabla f(x)\\cdot \\mathbf{u}$.\n\n- Differentiability: linear map approximating.\n\n- Tangent plane: $f(x)=f(x_0)+\\nabla f(x_0)\\cdot(x-x_0)+o(|x-x_0|)$.\n\n- Hessian: matrix of second partial derivatives.\n\n- Chain rule for multivariable: composition.\n\n- Implicit differentiation.\n\n- Multiple integrals: double integrals: $\\iint f(x,y)dxdy$.\n\n- Change of variable: Jacobian.\n\n- Vector fields: gradient, divergence, curl.\n\n- Greens, Stokes, divergence theorem.\n\nOk.\n\nLet's produce final answer. We'll produce structured sections. Use rigorous notation.\n\nLet's produce final.<|end|><|start|>assistant<|channel|>final<|message|># Calculus \u2013 From limits to multivariable calculus  \n*(All notation is written in the standard mathematical style; parentheses, subscripts and superscripts are used consistently.)*  \n\n---\n\n## 1.  One\u2011variable limits  \n\nLet \\(f:\\mathbb R\\to\\mathbb R\\).  \nFor \\(a\\in\\mathbb R\\) we say  \n\n\\[\n\\lim_{x\\to a}f(x)=L\n\\]\n\niff for every \\(\\varepsilon>0\\) there exists \\(\\delta>0\\) such that  \n\n\\[\n0<|x-a|<\\delta \\ \\Longrightarrow\\ |f(x)-L|<\\varepsilon .\n\\]\n\nThe same definition works for one\u2011dimensional limits at infinity or extended reals.\n\n**Continuity**  \n\\(f\\) is continuous at \\(a\\) iff  \n\n\\[\n\\lim_{x\\to a}f(x)=f(a).\n\\]\n\n---\n\n## 2.  Differentiation in one variable  \n\nThe derivative of \\(f\\) at \\(a\\) is defined by\n\n\\[\nf'(a)=\\lim_{h\\to0}\\frac{f(a+h)-f(a)}{h}\\qquad(\\text{provided the limit exists}).\n\\]\n\nIf \\(f'\\) exists on an interval, \\(f\\) is said to be differentiable there.\n\n**Basic rules**\n\n* Sum: \\((f+g)'=f'+g')\\)  \n* Product: \\((fg)'=f'g+fg'\\)  \n* Quotient: \\(\\left(\\frac{f}{g}\\right)'=\\frac{g f'-f g'}{g^2}\\)  \n* Chain: \\((f\\!\\circ\\!g)'=f'(g)\\,g')\\)  \n\n**Fundamental theorem of calculus**\n\n\\[\n\\int_a^bf'(x)\\,dx=f(b)-f(a).\n\\]\n\n---\n\n## 3.  Integration  \n\n### 3.1  Riemann Sum  \n\nFor a bounded function \\(f\\) on \\([a,b]\\)) define a partition  \n\\(P=\\{x_0,\\dots ,x_n\\}\\) with \\(x_0=a,x_n=b\\).  \nFor each sub\u2013interval \\([x_{i-1},x_i])\\) choose a tag \\(c_i\\).  \nThe Riemann sum is\n\n\\[\nS(f,P)=\\sum_{i=1}^n f(c_i)(x_i-x_{i-1}).\n\\]\n\nIf \\(\\lim_{\\|P\\|\\to0}S(f,P)\\) exists, its value is called  \n\\(\\int_a^b f(x)\\,dx\\).\n\n### 3.2  Fundamental theorem of integration\n\nIf \\(F\\) is an antiderivative of \\(f\\) on \\([a,b]\\)), i.e.  \n\\(F'=f\\), then\n\n\\[\n\\int_a^b f(x)\\,dx=F(b)-F(a).\n\\]\n\n---\n\n## 4.  Multivariable calculus  \n\nLet \\(f:\\mathbb R^n\\to\\mathbb R\\).  \nWe write \\(x=(x_1,\\dots ,x_n)\\).\n\n### 4.1  Partial derivatives  \n\nThe \\(i\\)-th partial derivative of \\(f\\) at \\(x\\) is\n\n\\[\n\\frac{\\partial f}{\\partial x_i}(x)=\n\\lim_{h\\to0}\\frac{f(x_1,\\dots ,x_i+h,\\dots ,x_n)-f(x)}{h}.\n\\]\n\nIf all partials exist in a neighbourhood of \\(x_0\\) and are continuous there, \\(f\\) is said to be **continuously differentiable** at \\(x_0\\).\n\n### 4.2  Gradient and directional derivative  \n\nThe gradient of \\(f\\) at \\(x\\) is the vector\n\n\\[\n\\nabla f(x)=\\Big(\\frac{\\partial f}{\\partial x_1}(x),\\dots ,\\frac{\\partial f}{\\partial x_n}(x)\\Big).\n\\]\n\nFor a unit vector \\(u=(u_1,\\dots ,u_n)\\) the directional derivative of \\(f\\) in direction \\(u\\) at \\(x\\) is\n\n\\[\nD_u f(x)=\\nabla f(x)\\cdot u\n=\\sum_{i=1}^n \\frac{\\partial f}{\\partial x_i}(x)u_i .\n\\]\n\n### 4.3  Tangent plane  \n\nIf \\(f\\) is differentiable at \\(x_0\\) then\n\n\\[\nf(x)=f(x_0)+\\nabla f(x_0)\\cdot(x-x_0)+o(|x-x_0|).\n\\]\n\nThe linear part \\( \\nabla f(x_0)\\cdot(x-x_0)\\) gives the equation of the tangent plane to the graph of \\(f\\) at \\((x_0,f(x_0))\\).\n\n### 4.4  Hessian  \n\nThe Hessian matrix of \\(f\\) at \\(x\\) is the \\(n\\times n\\) matrix\n\n\\[\nH_f(x)_{ij}=\\frac{\\partial^2 f}{\\partial x_i\\partial x_j}(x) .\n\\]\n\nIf \\(H_f(x)\\) is positive definite, \\(f\\) has a strict local minimum at \\(x\\); if negative definite, a strict local maximum; if indefinite, a saddle point.\n\n### 4.5  Chain rule in several variables  \n\nLet \\(g:\\mathbb R^m\\to\\mathbb R^n\\) and \\(h:\\mathbb R^k\\to\\mathbb R^m\\).  \nIf \\(g\\) and \\(h\\) are differentiable, then \\(g\\!\\circ\\!h\\) is differentiable and\n\n\\[\nD(g\\!\\circ\\!h)(x)=Dg(h(x))\\cdot Dh(x).\n\\]\n\n### 4.6  Implicit differentiation  \n\nSuppose \\(F(x_1,\\dots ,x_n,y)=0\\) defines \\(y\\) implicitly as a function of \\(x_1,\\dots ,x_n\\).  \nProvided \\(\\frac{\\partial F}{\\partial y}\\neq0\\) at a point, the implicit function theorem gives\n\n\\[\n\\frac{\\partial y}{\\partial x_i}\n= -\\,\\frac{\\frac{\\partial F}{\\partial x_i}}{\\frac{\\partial F}{\\partial y}}\\quad(i=1,\\dots ,n).\n\\]\n\n---\n\n## 5.  Multiple integrals  \n\nLet \\(f:\\mathbb R^2\\to\\mathbb R\\).  \nThe double integral over a region \\(R\\subseteq\\mathbb R^2\\) is\n\n\\[\n\\iint_R f(x,y)\\,dx\\,dy .\n\\]\n\nIf \\(R=\\{(x,y):a\\le x\\le b,\\ c(x)\\le y\\le d(x)\\}\\) then\n\n\\[\n\\iint_R f(x,y)\\,dx\\,dy\n=\\int_a^b\\Big(\\int_{c(x)}^{d(x)}f(x,y)\\,dy\\Big)\\,dx .\n\\]\n\nThe same pattern extends to triple and higher\u2011order integrals.\n\n---\n\n## 6.  Change\u2011of\u2011variables for multiple integrals  \n\nFor a one\u2011to\u2011one \\(C^1\\) map \\(T:\\mathbb R^n\\to\\mathbb R^n\\),\n\\(T=(T_1,\\dots ,T_n)\\) with Jacobian determinant \\(J_T(x)=\\det(DT(x))\\neq0\\),  \n\n\\[\n\\iint_{T(A)} f(u)\\,du\n=\\iint_A f(T(x))\\,|J_T(x)|\\,dx .\n\\]\n\n---\n\n## 7.  Vector fields, divergence, curl, and theorems  \n\nLet \\(F:\\mathbb R^3\\to\\mathbb R^3\\) be a smooth vector field.  \nFor a smooth orientable surface \\(S\\) with outward normal \\(n\\) we have\n\n* **Divergence theorem**  \n\n\\[\n\\iint_S(\\nabla\\cdot F)\\,dS=\\iint_{\\partial S}F\\cdot n,\\ dS .\n\\]\n\n* **Curl (rotation)**  \n\n\\[\n\\nabla\\times F=\\Big(\\frac{\\partial F_3}{\\partial y}-\\frac{\\partial F_2}{\\partial z},\\dots \\Big).\n\\]\n\n* **Stokes theorem**  \n\n\\[\n\\iint_{\\partial S}F\\cdot d\\ell=\\iint_S(\\nabla\\times F)\\cdot n,\\ dS .\n\\]\n\n---\n\n## 8.  Summary of key formulas  \n\n| Concept | Notation | Formula |\n|---|---|---|\n| Limit | \\(\\lim_{x\\to a}f(x)\\) | Definition with \\(\\varepsilon,\\delta\\) |\n| Derivative | \\(f'(a)\\) | \\(\\lim_{h\\to0}\\frac{f(a+h)-f(a)}{h}\\) |\n| Partial derivative | \\(\\frac{\\partial f}{\\partial x_i}\\) | \\(\\lim_{h\\to0}\\frac{f(...,x_i+h,...) -f(...)}{h}\\) |\n| Gradient | \\(\\nabla f\\) | \\((\\frac{\\partial f}{\\partial x_1},\\dots ,\\frac{\\partial f}{\\partial x_n})\\) |\n| Tangent plane | \\(f(x)\\) near \\(x_0\\) | \\(f(x)=f(x_0)+\\nabla f(x_0)\\cdot(x-x_0)+o(|x-x_0|)\\) |\n| Hessian | \\(H_f(x)\\) | \\(\\frac{\\partial^2 f}{\\partial x_i\\partial x_j}\\) |\n| Fundamental theorem of calculus | \\(\\int_a^bf'(x)\\,dx\\) | \\(f(b)-f(a)\\) |\n| Multiple integral | \\(\\iint_R f(x)\\,dx\\) | \\(\\iint_R f(x)\\,dx\\) (iterated integrals) |\n\n---\n\n###  The End\n\nThis overview gives the major rigorous notions and formulas that underlie calculus, from the first idea of limits to the breadth of multivariable calculus.  All expressions have been presented in precise mathematical notation, and each result is stated exactly as it appears in analytic theory."}
{"model": "mlx-community/gpt-oss-20b-MXFP4-Q8", "status": "OK", "load_time_s": 3, "gen_time_s": 40.89, "prompt_tokens": 33, "completion_tokens": 3284, "total_tokens": 3317, "tokens_per_sec": 80.31, "content_preview": "<|channel|>analysis<|message|>The user wants an explanation of calculus concepts from limits through multivariable calculus with rigorous mathematical notation. We need to provide a comprehensive but concise overview. Use LaTeX-like notation. Cover limits, continuity, derivative, integration, sequen", "content": "<|channel|>analysis<|message|>The user wants an explanation of calculus concepts from limits through multivariable calculus with rigorous mathematical notation. We need to provide a comprehensive but concise overview. Use LaTeX-like notation. Cover limits, continuity, derivative, integration, sequences, series, multivariable limits, partial derivatives, gradient, divergence, curl, multiple integrals, change of variables, differential forms maybe. Also mention rigorous definitions, epsilon-delta. Provide examples. Ensure mathematical notation is used. The user wants explanation of concepts, so maybe a structured outline. Probably use sections: 1) Limits, 2) Continuity, 3) Derivatives, 4) Integrals, 5) Fundamental theorem, 6) Sequences and series, 7) Multivariable limits, 8) Partial derivatives, 9) Gradient, 10) Divergence, 11) Curl, 12) Multiple integrals, 13) Change of variables, 14) Advanced topics. Provide definitions, theorems, proofs sketch. Use rigorous math. Let's produce a well-structured answer.<|end|><|start|>assistant<|channel|>final<|message|>Below is a concise yet rigorous tour of the main ideas that build classical calculus, from one\u2011variable limits up to the core of multivariable analysis.  All the key definitions are written in precise mathematical notation, and the most important theorems are stated (sometimes with an outline of the proof).  The goal is to give you a \u201cformula sheet\u201d that you can consult whenever you need a reference for a concept or a proof.\n\n---\n\n## 1. Limits of Functions (One Variable)\n\n**Definition (Limit).**  \nLet \\(f : D \\subset \\mathbb{R}\\to\\mathbb{R}\\) and let \\(a\\in \\mathbb{R}\\) be a *limit point* of \\(D\\).  \nWe say that\n\\[\n\\lim_{x\\to a} f(x)=L\n\\]\nif for every \\(\\varepsilon>0\\) there exists \\(\\delta>0\\) such that\n\\[\n0<|x-a|<\\delta \\;\\Longrightarrow\\; |f(x)-L|<\\varepsilon,\\qquad x\\in D .\n\\]\nNotation: \\(f(x)\\xrightarrow[x\\to a]{}L\\).\n\n**Uniqueness.**  \nIf both \\(L\\) and \\(M\\) satisfy the above, then \\(L=M\\).  \n*Sketch:* Suppose \\(\\varepsilon<|L-M|/2\\). Then there are \\(\\delta_1,\\delta_2\\) making the inequalities hold. Take \\(\\delta=\\min(\\delta_1,\\delta_2)\\); a contradiction arises unless \\(L=M\\).\n\n**Two\u2011sided and one\u2011sided limits.**  \n\\[\n\\lim_{x\\to a^-}f(x)=\\lim_{x\\to a^+}f(x)=L\n\\quad\\Longleftrightarrow\\quad\n\\lim_{x\\to a} f(x)=L .\n\\]\n\n**Algebra of limits.**  \nIf \\(\\lim_{x\\to a} f(x)=L\\) and \\(\\lim_{x\\to a} g(x)=M\\) then\n\\[\n\\begin{aligned}\n\\lim_{x\\to a}(f+g)(x) &= L+M,\\\\\n\\lim_{x\\to a}(f\\cdot g)(x) &= LM,\\\\\n\\lim_{x\\to a}\\frac{f}{g}(x) &= \\frac{L}{M}\\quad(M\\neq 0).\n\\end{aligned}\n\\]\n\n**Squeeze theorem.**  \nIf \\(h(x)\\le f(x)\\le g(x)\\) for all \\(x\\) near \\(a\\) and \\(\\lim_{x\\to a}h(x)=\\lim_{x\\to a}g(x)=L\\), then \\(\\lim_{x\\to a}f(x)=L\\).\n\n---\n\n## 2. Continuity and Uniform Continuity\n\n**Definition (Continuity at a point).**  \n\\(f\\) is continuous at \\(a\\) if\n\\[\n\\lim_{x\\to a} f(x)=f(a).\n\\]\n\n**Definition (Continuity on a set).**  \n\\(f\\) is continuous on \\(E\\subset\\mathbb{R}\\) if it is continuous at every \\(a\\in E\\).\n\n**Uniform Continuity.**  \n\\(f:E\\to\\mathbb{R}\\) is uniformly continuous if for every \\(\\varepsilon>0\\) there exists \\(\\delta>0\\) such that\n\\[\n|x-y|<\\delta \\;\\Longrightarrow\\; |f(x)-f(y)|<\\varepsilon,\\qquad \\forall x,y\\in E .\n\\]\nUniform continuity is *independent* of the particular point \\(a\\).\n\n**Heine\u2013Cantor Theorem.**  \nIf \\(E\\) is a compact subset of \\(\\mathbb{R}\\) and \\(f:E\\to\\mathbb{R}\\) is continuous, then \\(f\\) is uniformly continuous on \\(E\\).\n\n---\n\n## 3. Sequences, Series, and Convergence\n\n### 3.1 Sequences\n\n**Definition (Limit of a sequence).**  \nA sequence \\((a_n)_{n\\ge 1}\\subset\\mathbb{R}\\) converges to \\(L\\) if\n\\[\n\\forall \\varepsilon>0\\;\\exists N\\in\\mathbb{N}\\;\\forall n\\ge N:\\; |a_n-L|<\\varepsilon .\n\\]\nNotation: \\(a_n\\xrightarrow[n\\to\\infty]{}L\\).\n\n**Cauchy Criterion.**  \n\\((a_n)\\) converges iff for every \\(\\varepsilon>0\\) there exists \\(N\\) such that\n\\[\nm,n\\ge N\\;\\Longrightarrow\\; |a_n-a_m|<\\varepsilon .\n\\]\n\n### 3.2 Series\n\n**Definition (Infinite series).**  \nGiven \\((a_n)\\), the series \\(\\sum_{n=1}^{\\infty} a_n\\) converges to \\(S\\) iff the sequence of partial sums\n\\[\nS_N=\\sum_{n=1}^{N} a_n\n\\]\nconverges to \\(S\\).\n\n**Absolute convergence.**  \nIf \\(\\sum_{n=1}^{\\infty} |a_n|\\) converges, then \\(\\sum a_n\\) converges (absolute convergence \u21d2 convergence).\n\n**Comparison Test.**  \nIf \\(0\\le a_n\\le b_n\\) for all \\(n\\) and \\(\\sum b_n\\) converges, then \\(\\sum a_n\\) converges.\n\n**Ratio Test.**  \nIf \\(\\lim_{n\\to\\infty} \\left|\\frac{a_{n+1}}{a_n}\\right|=L\\) exists, then the series converges absolutely if \\(L<1\\) and diverges if \\(L>1\\). (The case \\(L=1\\) is inconclusive.)\n\n**Integral Test.**  \nIf \\(f:[1,\\infty)\\to[0,\\infty)\\) is continuous, decreasing, and \\(f(n)=a_n\\), then\n\\[\n\\sum_{n=1}^{\\infty} a_n \\text{ converges }\\iff\\int_1^{\\infty} f(x)\\,dx \\text{ converges}.\n\\]\n\n---\n\n## 4. Differentiation (One Variable)\n\n### 4.1 Definition\n\nLet \\(f:D\\to\\mathbb{R}\\) and \\(a\\in D\\) be an interior point.  \n\\(f\\) is differentiable at \\(a\\) if there exists a real number \\(f'(a)\\) such that\n\\[\n\\lim_{h\\to 0}\\frac{f(a+h)-f(a)}{h}=f'(a).\n\\]\nThe quantity \\(f'(a)\\) is the *derivative* of \\(f\\) at \\(a\\).\n\n### 4.2 Rules of Differentiation\n\nIf \\(f,g\\) are differentiable at \\(a\\), then\n\n| Operation | Derivative |\n|---|---|\n| Sum | \\( (f+g)'(a)=f'(a)+g'(a)\\) |\n| Difference | \\( (f-g)'(a)=f'(a)-g'(a)\\) |\n| Product | \\( (fg)'(a)=f'(a)g(a)+f(a)g'(a)\\) |\n| Quotient | \\( \\left(\\frac{f}{g}\\right)'(a)=\\frac{f'(a)g(a)-f(a)g'(a)}{g(a)^2}\\) |\n| Chain rule | If \\(g\\) is differentiable at \\(a\\) and \\(f\\) at \\(g(a)\\), then \\((f\\circ g)'(a)=f'(g(a))\\,g'(a)\\) |\n\n### 4.3 Mean Value Theorem (MVT)\n\nLet \\(f:[a,b]\\to\\mathbb{R}\\) be continuous on \\([a,b]\\) and differentiable on \\((a,b)\\). Then there exists \\(c\\in(a,b)\\) such that\n\\[\nf'(c)=\\frac{f(b)-f(a)}{b-a}.\n\\]\n\n### 4.4 Taylor\u2019s Theorem\n\nIf \\(f\\) has continuous derivatives up to order \\(n\\) on an interval containing \\(a\\) and \\(x\\), then\n\\[\nf(x)=\\sum_{k=0}^{n}\\frac{f^{(k)}(a)}{k!}(x-a)^k+R_n(x),\n\\]\nwhere the remainder satisfies the Lagrange form\n\\[\nR_n(x)=\\frac{f^{(n+1)}(\\xi)}{(n+1)!}(x-a)^{\\,n+1}\\quad\\text{for some }\\xi\\text{ between }a\\text{ and }x .\n\\]\n\n---\n\n## 5. Integration (One Variable)\n\n### 5.1 Riemann Integral\n\nLet \\(f:[a,b]\\to\\mathbb{R}\\). A *partition* \\(P\\) of \\([a,b]\\) is a finite sequence\n\\[\nP:\\ a=x_0<x_1<\\dots<x_N=b .\n\\]\nDefine \\(\\Delta x_i=x_i-x_{i-1}\\) and\n\\[\nL(f,P)=\\sum_{i=1}^{N}\\inf_{x\\in[x_{i-1},x_i]}f(x)\\,\\Delta x_i,\\qquad\nU(f,P)=\\sum_{i=1}^{N}\\sup_{x\\in[x_{i-1},x_i]}f(x)\\,\\Delta x_i.\n\\]\nIf \\(\\sup_{P}L(f,P)=\\inf_{P}U(f,P)=I\\), we say \\(f\\) is **Riemann integrable** on \\([a,b]\\) and write\n\\[\n\\int_{a}^{b}f(x)\\,dx=I .\n\\]\n\n### 5.2 Fundamental Theorem of Calculus (FTC)\n\n**Part\u202f1.** If \\(f\\) is integrable on \\([a,b]\\) and \\(F(x)=\\int_{a}^{x}f(t)\\,dt\\), then \\(F\\) is continuous on \\([a,b]\\) and differentiable on \\((a,b)\\) with \\(F'(x)=f(x)\\).\n\n**Part\u202f2.** If \\(f\\) is continuous on \\([a,b]\\) and \\(F\\) is any antiderivative of \\(f\\) on \\([a,b]\\), then\n\\[\n\\int_{a}^{b}f(x)\\,dx=F(b)-F(a).\n\\]\n\n### 5.3 Change of Variables\n\nIf \\(\\varphi:[\\alpha,\\beta]\\to[a,b]\\) is continuously differentiable and one\u2011to\u2011one, then\n\\[\n\\int_{a}^{b}f(x)\\,dx=\\int_{\\alpha}^{\\beta}f(\\varphi(t))\\,\\varphi'(t)\\,dt .\n\\]\n\n---\n\n## 6. Multivariable Limits\n\nLet \\(f:\\mathbb{R}^n\\supset D\\to\\mathbb{R}\\) and \\(a\\in\\overline{D}\\) a limit point.\n\n**Definition (Limit).**  \n\\[\n\\lim_{\\mathbf{x}\\to\\mathbf{a}}f(\\mathbf{x})=L\n\\]\niff for every \\(\\varepsilon>0\\) there exists \\(\\delta>0\\) such that\n\\[\n0<\\|\\mathbf{x}-\\mathbf{a}\\|<\\delta \\;\\Longrightarrow\\; |f(\\mathbf{x})-L|<\\varepsilon .\n\\]\n\n**Remark.**  \nIn \\(\\mathbb{R}^n\\) the squeeze theorem still works, but one must check limits along *all* paths; a counterexample is \\(f(x,y)=\\frac{xy}{x^2+y^2}\\) at \\((0,0)\\).\n\n---\n\n## 7. Partial Derivatives and Differentiability\n\nLet \\(f:\\mathbb{R}^n\\supset D\\to\\mathbb{R}\\). For \\(i=1,\\dots,n\\),\n\n**Partial derivative w.r.t. \\(x_i\\):**\n\\[\n\\frac{\\partial f}{\\partial x_i}(\\mathbf{a})=\n\\lim_{h\\to0}\\frac{f(a_1,\\dots,a_i+h,\\dots,a_n)-f(\\mathbf{a})}{h},\n\\]\nprovided the limit exists.\n\n**Differentiability.**  \n\\(f\\) is differentiable at \\(\\mathbf{a}\\) if there exists a linear map \\(L:\\mathbb{R}^n\\to\\mathbb{R}\\) such that\n\\[\n\\lim_{\\mathbf{h}\\to\\mathbf{0}}\\frac{f(\\mathbf{a}+\\mathbf{h})-f(\\mathbf{a})-L(\\mathbf{h})}{\\|\\mathbf{h}\\|}=0.\n\\]\nIf differentiable, the matrix of \\(L\\) is the gradient\n\\[\n\\nabla f(\\mathbf{a})=\n\\left(\\frac{\\partial f}{\\partial x_1}(\\mathbf{a}),\\dots,\n\\frac{\\partial f}{\\partial x_n}(\\mathbf{a})\\right).\n\\]\n\n**Chain Rule (Multivariable).**  \nIf \\(\\mathbf{g}:\\mathbb{R}^m\\to\\mathbb{R}^n\\) and \\(f:\\mathbb{R}^n\\to\\mathbb{R}\\) are differentiable, then\n\\[\n\\frac{d}{dx_j}\\bigl(f\\circ\\mathbf{g}\\bigr)(\\mathbf{x})=\n\\sum_{i=1}^{n}\\frac{\\partial f}{\\partial y_i}(\\mathbf{g}(\\mathbf{x}))\n\\frac{\\partial g_i}{\\partial x_j}(\\mathbf{x}).\n\\]\n\n---\n\n## 8. Vector Calculus Identities\n\nLet \\(\\mathbf{F}:\\mathbb{R}^3\\to\\mathbb{R}^3\\) be continuously differentiable.\n\n| Symbol | Definition | Interpretation |\n|---|---|---|\n| \\(\\nabla\\) | \\(\\displaystyle\\nabla=\\left(\\frac{\\partial}{\\partial x},\n\\frac{\\partial}{\\partial y},\n\\frac{\\partial}{\\partial z}\\right)\\) | Gradient operator |\n| \\(\\nabla\\cdot\\mathbf{F}\\) | \\(\\displaystyle\\frac{\\partial F_x}{\\partial x}\n+\\frac{\\partial F_y}{\\partial y}\n+\\frac{\\partial F_z}{\\partial z}\\) | Divergence |\n| \\(\\nabla\\times\\mathbf{F}\\) | \\(\\displaystyle\n\\left(\\frac{\\partial F_z}{\\partial y}-\\frac{\\partial F_y}{\\partial z},\n\\frac{\\partial F_x}{\\partial z}-\\frac{\\partial F_z}{\\partial x},\n\\frac{\\partial F_y}{\\partial x}-\\frac{\\partial F_x}{\\partial y}\\right)\\) | Curl |\n| \\(\\Delta f\\) | \\(\\displaystyle\\nabla\\cdot\\nabla f=\n\\frac{\\partial^2 f}{\\partial x^2}\n+\\frac{\\partial^2 f}{\\partial y^2}\n+\\frac{\\partial^2 f}{\\partial z^2}\\) | Laplacian |\n\n**Fundamental Theorems**\n\n1. *Gradient Theorem (Fundamental Theorem of Line Integrals).*\n   For a smooth curve \\(\\gamma:[a,b]\\to\\mathbb{R}^n\\),\n   \\[\n   \\int_{\\gamma}\\nabla f\\cdot d\\mathbf{r}\n   =f(\\gamma(b))-f(\\gamma(a)).\n   \\]\n2. *Divergence Theorem (Gauss).*\n   If \\(V\\subset\\mathbb{R}^3\\) has piecewise smooth boundary \\(\\partial V\\) oriented outward,\n   \\[\n   \\int_{V}\\nabla\\cdot\\mathbf{F}\\,dV\n   =\\int_{\\partial V}\\mathbf{F}\\cdot\\mathbf{n}\\,dS .\n   \\]\n3. *Stokes\u2019 Theorem.*\n   For a smooth oriented surface \\(S\\) with boundary curve \\(\\partial S\\),\n   \\[\n   \\int_{S}\\nabla\\times\\mathbf{F}\\cdot d\\mathbf{S}\n   =\\int_{\\partial S}\\mathbf{F}\\cdot d\\mathbf{r}.\n   \\]\n\n---\n\n## 9. Multiple (Repeated) Integrals\n\nLet \\(f:\\mathbb{R}^n\\to\\mathbb{R}\\) be integrable over a rectangular region\n\\(R=[a_1,b_1]\\times\\cdots\\times[a_n,b_n]\\).\n\n**Definition (Iterated Integral).**  \n\\[\n\\int_{R} f(\\mathbf{x})\\,d\\mathbf{x}\n=\n\\int_{a_n}^{b_n}\\!\\!\\cdots\\int_{a_1}^{b_1}\nf(x_1,\\dots,x_n)\\,dx_1\\cdots dx_n .\n\\]\n\n**Fubini\u2019s Theorem.**  \nIf \\(f\\) is integrable on \\(R\\), the order of integration may be interchanged:\n\\[\n\\int_{R} f\\,d\\mathbf{x}\n=\n\\int_{a_{\\sigma(1)}}^{b_{\\sigma(1)}}\\!\\!\\cdots\n\\int_{a_{\\sigma(n)}}^{b_{\\sigma(n)}}\nf\\bigl(x_{\\sigma(1)},\\dots,x_{\\sigma(n)}\\bigr)\n\\,dx_{\\sigma(1)}\\cdots dx_{\\sigma(n)} .\n\\]\n\n**Change of Variables (Jacobian).**  \nLet \\(\\Phi:\\Omega\\subset\\mathbb{R}^n\\to\\mathbb{R}^n\\) be a \\(C^1\\) bijection with inverse \\(\\Psi\\). Then\n\\[\n\\int_{\\Phi(\\Omega)} f(\\mathbf{x})\\,d\\mathbf{x}\n=\n\\int_{\\Omega} f\\bigl(\\Phi(\\mathbf{u})\\bigr)\\,|\\det D\\Phi(\\mathbf{u})|\\,d\\mathbf{u},\n\\]\nwhere \\(D\\Phi\\) is the Jacobian matrix of \\(\\Phi\\).\n\n---\n\n## 10. Differential Forms (Optional, for completeness)\n\nA \\(k\\)-form \\(\\omega\\) on \\(\\mathbb{R}^n\\) is a linear combination of elements\n\\(dx_{i_1}\\wedge\\cdots\\wedge dx_{i_k}\\).  \nThe exterior derivative \\(d\\) satisfies \\(d^2=0\\) and generalizes gradient, curl, and divergence:\n\\[\n\\begin{aligned}\nd(f)&=\\nabla f\\cdot d\\mathbf{r},\\\\\nd(\\mathbf{F}\\cdot d\\mathbf{r})&=(\\nabla\\times\\mathbf{F})\\cdot d\\mathbf{S},\\\\\nd(\\mathbf{F}\\cdot d\\mathbf{S})&=(\\nabla\\cdot\\mathbf{F})\\,dV .\n\\end{aligned}\n\\]\nStokes\u2019 theorem in this language reads \\(\\int_{\\partial M}\\omega = \\int_M d\\omega\\).\n\n---\n\n## 11. Common Pitfalls & Key Takeaways\n\n| Issue | Remedy |\n|---|---|\n| Assuming a limit exists because \u201cboth approaches give the same number.\u201d | Verify with the \\(\\varepsilon\\)-\\(\\delta\\) definition. |\n| Differentiating \u201cpiecewise\u2011defined\u201d functions without checking differentiability at junctions. | Check continuity of derivatives or apply the one\u2011sided derivative test. |\n| Confusing the domain of a function with its range. | Explicitly state \\(f:D\\to\\mathbb{R}\\) and \\(\\operatorname{Im}(f)\\). |\n| Using the Fundamental Theorem of Calculus for a function that is only integrable, not continuous. | Require continuity on \\([a,b]\\) for FTC part\u202f2. |\n| Applying the divergence theorem without orienting the boundary correctly. | Ensure the outward normal convention is used. |\n\n---\n\n### Final Thought\n\nThe structure we have sketched is the backbone of classical analysis.  Each definition is carefully crafted to guarantee that the theorems we prove later\u2014limits, continuity, differentiation, integration, and their multivariable counterparts\u2014work in a *uniform* way across all contexts.  Keeping the notation tight and the hypotheses exact is the key to mastering calculus at a rigorous level."}
{"model": "mlx-community/granite-4.0-h-tiny-4bit", "status": "OK", "load_time_s": 2, "gen_time_s": 8.3, "prompt_tokens": 27, "completion_tokens": 982, "total_tokens": 1009, "tokens_per_sec": 118.29, "content_preview": "Certainly! Calculus is a fundamental branch of mathematics that deals with the study of change and motion. It provides a framework for modeling and analyzing dynamic systems. We'll start with the concept of limits and gradually progress towards multivariable calculus.  ### Limits  Limits are the fou", "content": "Certainly! Calculus is a fundamental branch of mathematics that deals with the study of change and motion. It provides a framework for modeling and analyzing dynamic systems. We'll start with the concept of limits and gradually progress towards multivariable calculus.\n\n### Limits\n\nLimits are the foundation of calculus. They describe the value that a function approaches as the input (or variable) approaches a certain point. Formally, the limit of a function \\(f(x)\\) as \\(x\\) approaches \\(a\\) is denoted as:\n\n\\[\n\\lim_{x \\to a} f(x) = L\n\\]\n\nThis means that for every \\(\\epsilon > 0\\), there exists a \\(\\delta > 0\\) such that for all \\(x\\), \\(0< |x - a|< \\delta\\) implies \\(|f(x) - L|< \\epsilon\\).\n\n### Continuity\n\nA function \\(f(x)\\) is continuous at a point \\(a\\) if \\(\\lim_{x \\to a} f(x) = f(a)\\). This implies that the function has no breaks, jumps, or holes at \\(x = a\\).\n\n### Derivatives\n\nThe derivative of a function at a point gives the slope of the tangent to the function at that point. It's defined as:\n\n\\[\nf'(x) = \\lim_{h \\to 0} \\frac{f(x + h) - f(x)}{h}\n\\]\n\nDerivatives are used to find the rate of change of functions. They are crucial for optimization problems, where we seek to find maxima and minima of functions.\n\n### Integrals\n\nIntegration is essentially the reverse process of differentiation. The definite integral of a function over an interval \\([a, b]\\) gives the signed area under the curve of the function between \\(a\\) and \\(b\\):\n\n\\[\n\\int_a^b f(x) \\, dx\n\\]\n\nIndefinite integrals represent a family of functions and are the antiderivatives of functions.\n\n### Multivariable Calculus\n\nWhen calculus is extended to functions of multiple variables, several new concepts and techniques emerge. Concepts like partial derivatives, multiple integrals, and vector calculus are central to this field.\n\n#### Partial Derivatives\n\nFor a function \\(f(x, y)\\) of two variables, the partial derivative with respect to \\(x\\) is defined as:\n\n\\[\n\\frac{\\partial f}{\\partial x} = \\lim_{h \\to 0} \\frac{f(x + h, y) - f(x, y)}{h}\n\\]\n\nSimilarly, the partial derivative with respect to \\(y\\) is:\n\n\\[\n\\frac{\\partial f}{\\partial y} = \\lim_{h \\to 0} \\frac{f(x, y + h) - f(x, y)}{h}\n\\]\n\n#### Multiple Integrals\n\nMultiple integrals extend the concept of integrals to functions of multiple variables. For example, a double integral over a region \\(D\\) in the \\(xy\\)-plane is:\n\n\\[\n\\iint_D f(x, y) \\, dA = \\iint_D f(x, y) \\, dx \\, dy\n\\]\n\nwhere \\(dA\\) represents an infinitesimal area element.\n\n#### Gradient, Divergence, and Curl\n\nIn vector calculus, these operations are defined for vector fields (functions that assign a vector to each point in space).\n\n- **Gradient**: For a scalar field \\(f(x, y, z)\\), the gradient is a vector field:\n  \\[\n  \\nabla f = \\left( \\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial z} \\right)\n  \\]\n\n- **Divergence**: For a vector field \\(\\mathbf{F} = (F_x, F_y, F_z)\\), the divergence is a scalar:\n  \\[\n  \\nabla \\cdot \\mathbf{F} = \\frac{\\partial F_x}{\\partial x} + \\frac{\\partial F_y}{\\partial y} + \\frac{\\partial F_z}{\\partial z}\n  \\]\n\n- **Curl**: The curl of a vector field \\(\\mathbf{F}\\) is a vector field:\n  \\[\n  \\nabla \\times \\mathbf{F} = \\left( \\frac{\\partial F_z}{\\partial y} - \\frac{\\partial F_y}{\\partial z}, \\frac{\\partial F_x}{\\partial z} - \\frac{\\partial F_z}{\\partial x}, \\frac{\\partial F_y}{\\partial x} - \\frac{\\partial F_x}{\\partial y} \\right)\n  \\]\n\nThese concepts are fundamental in physics, particularly in electromagnetism and fluid dynamics.\n\n### Conclusion\n\nCalculus, from its foundational concepts of limits and derivatives to its extension into multivariable calculus with integrals, partial derivatives, and vector calculus, provides powerful tools for analyzing and modeling a wide range of phenomena in the physical world. Understanding these concepts requires rigorous mathematical notation and a solid grasp of the underlying principles."}
{"model": "mlx-community/JoyAI-LLM-Flash-4bit-DWQ", "status": "OK", "load_time_s": 9, "gen_time_s": 16.86, "prompt_tokens": 27, "completion_tokens": 721, "total_tokens": 748, "tokens_per_sec": 42.76, "content_preview": "## **Calculus: From Limits to Multivariable Calculus**  ### **1. Limits & Continuity** $$ \\lim_{x \\to a} f(x) = L \\iff \\forall \\epsilon > 0, \\exists \\delta > 0: 0< |x-a|< \\delta \\implies |f(x)-L|< \\epsilon $$ $$ f \\text{ continuous at } a \\iff \\lim_{x \\to a} f(x) = f(a) $$  ### **2. Differentiation*", "content": "## **Calculus: From Limits to Multivariable Calculus**\n\n### **1. Limits & Continuity**\n$$\n\\lim_{x \\to a} f(x) = L \\iff \\forall \\epsilon > 0, \\exists \\delta > 0: 0< |x-a|< \\delta \\implies |f(x)-L|< \\epsilon\n$$\n$$\nf \\text{ continuous at } a \\iff \\lim_{x \\to a} f(x) = f(a)\n$$\n\n### **2. Differentiation**\n$$\nf'(x) = \\lim_{h \\to 0} \\frac{f(x+h)-f(x)}{h}\n$$\n$$\n\\text{Chain Rule: } \\frac{d}{dx} g(f(x)) = g'(f(x)) \\cdot f'(x)\n$$\n$$\n\\text{Product Rule: } (uv)' = u'v + uv'\n$$\n\n### **3. Integration**\n$$\n\\int_a^b f(x) dx = \\lim_{n \\to \\infty} \\sum_{i=1}^n f(x_i^*) \\Delta x_i \\quad (x_i^* \\in [x_{i-1}, x_i])\n$$\n$$\n\\text{FTC: } \\int_a^b f'(x) dx = f(b) - f(a)\n$$\n\n### **4. Series & Sequences**\n$$\na_n \\text{ converges to } L \\iff \\forall \\epsilon > 0, \\exists N: n > N \\implies |a_n - L|< \\epsilon\n$$\n$$\n\\text{Monotonic Convergence: } a_n \\uparrow L \\iff (a_n \\text{ increasing}) \\& ( \\sup a_n = L)\n$$\n\n### **5. Multivariable Calculus**\n#### **Partial Derivatives**\n$$\nf_x(x,y) = \\lim_{h \\to 0} \\frac{f(x+h,y)-f(x,y)}{h}\n$$\n$$\n\\text{Total Differential: } df = f_x dx + f_y dy + \\cdots\n$$\n\n#### **Multivariable Integration**\n$$\n\\iiint_R f(x,y,z) dx dy dz = \\lim_{\\Delta V \\to 0} \\sum_{cells} f(\\text{sample}) \\Delta V\n$$\n$$\n\\text{FTC in } \\mathbb{R}^n: \\int_{\\partial R} \\nabla f \\cdot d\\mathbf{S} = f|_{\\text{boundary}}\n$$\n\n#### **Taylor Series**\n$$\nf(\\mathbf{a}+\\mathbf{h}) = f(\\mathbf{a}) + \\nabla f(\\mathbf{a}) \\cdot \\mathbf{h} + \\frac{1}{2} \\mathbf{h}^T H(f(\\mathbf{a}) \\mathbf{h}) + \\cdots\n$$\n$$\n\\text{Where } H(f) = \\text{Hessian Matrix } [f_{ij}]\n$$\n\n#### **Vector Calculus**\n$$\n\\nabla f = \\text{gradient vector } (f_x, f_y, f_z)\n$$\n$$\n\\nabla \\cdot \\mathbf{F} = \\text{divergence } \\sum_i F_i\n$$\n$$\n\\nabla \\times \\mathbf{F} = \\text{curl } \\text{(cross product of operators)}\n$$\n\n### **6. Rigorous Foundations**\n$$\n\\text{Calculus } \\subset \\text{Analysis} \\subset \\text{Topology}\n$$\n$$\n\\text{Measure Theory } \\text{(Lebesgue Integration)} \\text{ extends Riemann Integration}\n$$\n$$\n\\text{Complex Analysis } \\text{(Continuity in } \\mathbb{C}) \\text{ differs from } \\mathbb{R}^n\n$$\n\n### **7. Key Theorems**\n$$\n\\text{Mean Value Theorem: } \\exists c \\in (a,b): f'(c) = \\frac{f(b)-f(a)}{b-a}\n$$\n$$\n\\text{Implicit Differentiation: } \\frac{dy}{dx} = -\\frac{F_x}{F_y} \\quad \\text{(where } F(x,y)=0)\n$$\n$$\n\\text{Jacobian Determinants: } J = \\text{det}[\\partial_i f_j] \\text{ for coordinate transforms}\n$$\n\n### **8. Advanced Topics**\n$$\n\\text{Stochastic Calculus: } dX_t = \\mu_t dt + \\sigma_t dW_t \\quad \\text{(Wiener Process)}\n$$\n$$\n\\text{Fourier Analysis: } \\hat{f}(\\omega) = \\int_{-\\infty}^{\\infty} f(t) e^{-i\\omega t} dt\n$$\n$$\n\\text{Differential Geometry: } \\nabla \\cdot \\nabla f = \\text{Laplacian } \\Delta f\n$$\n\n**This framework provides rigorous mathematical notation spanning from single-variable limits through to multivariable calculus, maintaining consistency across all calculus concepts.**"}
{"model": "mlx-community/LFM2-2.6B-4bit", "status": "OK", "load_time_s": 1, "gen_time_s": 9.34, "prompt_tokens": 27, "completion_tokens": 1741, "total_tokens": 1768, "tokens_per_sec": 186.35, "content_preview": "Certainly! Let's explore the progression of calculus concepts from **limits** to **multivariable calculus**, building rigorously from the foundations of single-variable calculus through to the multivariate setting with precise mathematical notation.  ---  ## 1. **Limits (Single-Variable Calculus)** ", "content": "Certainly! Let's explore the progression of calculus concepts from **limits** to **multivariable calculus**, building rigorously from the foundations of single-variable calculus through to the multivariate setting with precise mathematical notation.\n\n---\n\n## 1. **Limits (Single-Variable Calculus)**\n\nLimits are foundational in calculus and form the basis for continuity, derivatives, and integrals.\n\n### Definition (Single-Variable Limit):\nWe say that  \n\\[\n\\lim_{x \\to a} f(x) = L\n\\]\nexists and equals \\(L\\) if for every \\(\\varepsilon > 0\\), there exists \\(\\delta > 0\\) such that:\n\\[\n\\text{If } 0< |x - a|< \\delta, \\text{ then } |f(x) - L|< \\varepsilon.\n\\]\n\nThis definition ensures that \\(f(x)\\) approaches \\(L\\) as \\(x \\to a\\), no matter how close \\(x\\) gets to \\(a\\) (excluding \\(x = a\\) itself).\n\n---\n\n## 2. **Improper Limits and Infinite Limits**\n\nLimits can also involve infinite behavior:\n\n\\[\n\\lim_{x \\to a} f(x) = \\infty \\quad \\text{or} \\quad -\\infty \\quad \\text{if } |f(x)| > M \\text{ for all } |x - a|< \\delta.\n\\]\n\n---\n\n## 3. **Derivatives (Single-Variable Analysis)**\n\nThe derivative of \\(f\\) at \\(a\\) is defined as:\n\\[\nf'(a) = \\lim_{h \\to 0} \\frac{f(a + h) - f(a)}{h},\n\\]\nprovided the limit exists.\n\nThis defines the instantaneous rate of change and the slope of the tangent line.\n\n---\n\n## 4. **Partial Derivatives (Multivariable Calculus Preview)**\n\nWhen moving to functions of multiple variables, we generalize the concept of derivatives.\n\nLet \\(f: \\mathbb{R}^n \\to \\mathbb{R}\\). The **partial derivative** of \\(f\\) with respect to \\(x_i\\) at point \\((a_1, \\dots, a_n)\\) is defined as:\n\\[\n\\frac{\\partial f}{\\partial x_i}(a) = \\lim_{h \\to 0} \\frac{f(a_1, \\dots, (a_i + h), \\dots, a_n) - f(a_1, \\dots, a_i, \\dots, a_n)}{h},\n\\]\nprovided the limit exists.\n\nNote: In multivariable calculus, differentials \\(df\\) and directional derivatives generalize this idea.\n\n---\n\n## 5. **Directional Derivatives and Gradient**\n\n### Directional Derivative:\nLet \\( \\mathbf{u} \\in \\mathbb{R}^n \\) be a unit vector. The directional derivative of \\(f\\) at \\(a\\) in the direction \\(\\mathbf{u}\\) is:\n\\[\nD_{\\mathbf{u}}f(a) = \\frac{f(a + \\mathbf{u}) - f(a)}{\\|\\mathbf{u}\\|} = \\nabla f(a) \\cdot \\mathbf{u},\n\\]\nwhere \\(\\nabla f(a) = \\left( \\frac{\\partial f}{\\partial x_1}(a), \\dots, \\frac{\\partial f}{\\partial x_n}(a) \\right)\\).\n\n### Gradient Vector:\nThe gradient \\(\\nabla f(a)\\) is the vector of partial derivatives:\n\\[\n\\nabla f(a) = \\left( \\frac{\\partial f}{\\partial x_1}(a), \\dots, \\frac{\\partial f}{\\partial x_n}(a) \\right).\n\\]\n\n---\n\n## 6. **Chain Rule (Multivariable)**\n\nIf \\( z = f(x, y) \\) and \\( x = g(t), y = h(t) \\), then:\n\\[\n\\frac{dz}{dt} = \\frac{\\partial f}{\\partial x} \\frac{dx}{dt} + \\frac{\\partial f}{\\partial y} \\frac{dy}{dt} = \\frac{\\partial f}{\\partial x} \\frac{dg}{dt} + \\frac{\\partial f}{\\partial y} \\frac{dh}{dt}.\n\\]\n\nThis generalizes the single-variable chain rule.\n\n---\n\n## 7. **Multiple Integrals (Double and Triple Integrals)**\n\nWe extend integration to higher dimensions.\n\n### Double Integral (Over a Rectangle):\n\\[\n\\iint_R f(x, y) \\, dA = \\lim_{m,n \\to \\infty} \\sum_{m=1}^\\infty \\sum_{n=1}^\\infty f(a_m, b_n) \\Delta A,\n\\]\nwhere \\(R = [a_1, b_1] \\times [a_2, b_2]\\), and \\(\\Delta A\\) is the area of each subrectangle.\n\n### Triple Integral (Over a Rectangular Region):\n\\[\n\\iiint_E f(x, y, z) \\, dV = \\lim_{m,n,p \\to \\infty} \\sum_{i=1}^m \\sum_{j=1}^n \\sum_{k=1}^p f(a_i, b_j, c_k) \\Delta A \\Delta B \\Delta C.\n\\]\n\n---\n\n## 8. **Partial Derivatives of Integrals**\n\nUsing Leibniz's Rule, the partial derivative of a multiple integral with respect to a parameter (say \\(a_1\\)):\n\n\\[\n\\frac{\\partial}{\\partial a_1} \\left( \\iint f(x, y, a_1) \\, dx \\, dy \\right) = \\iint \\frac{\\partial f}{\\partial a_1}(x, y, a_1) \\, dx \\, dy.\n\\]\n\nThis is valid when \\(f\\) is sufficiently smooth.\n\n---\n\n## 9. **Gradient of a Function of Multiple Variables**\n\nThe gradient of \\(f: \\mathbb{R}^n \\to \\mathbb{R}\\) at a point \\((a_1, \\dots, a_n)\\) is:\n\\[\n\\nabla f(a_1, \\dots, a_n) = \\left( \\frac{\\partial f}{\\partial x_1}(a), \\dots, \\frac{\\partial f}{\\partial x_n}(a) \\right).\n\\]\n\nThis vector points in the direction of maximum increase of \\(f\\).\n\n---\n\n## 10. **Directional Derivative in General Direction**\n\nGiven a direction \\(\\mathbf{u} = (u_1, \\dots, u_n)\\), not necessarily a unit vector:\n\n\\[\nD_{\\mathbf{u}} f(a) = \\nabla f(a) \\cdot \\mathbf{u} = \\sum_{i=1}^n \\frac{\\partial f}{\\partial x_i}(a) u_i.\n\\]\n\n---\n\n## 11. **Hessian Matrix (Second Derivatives)**\n\nAt a point \\(a\\), the **Hessian matrix** \\(H_f(a)\\) of \\(f: \\mathbb{R}^n \\to \\mathbb{R}\\) is the matrix of second partial derivatives:\n\\[\nH_f(a) = \\begin{bmatrix}\n\\frac{\\partial^2 f}{\\partial x_1^2}(a) & \\cdots & \\frac{\\partial^2 f}{\\partial x_1 \\partial x_n}(a) \\\\\n\\vdots & \\ddots & \\vdots \\\\\n\\frac{\\partial^2 f}{\\partial x_n \\partial x_1}(a) & \\cdots & \\frac{\\partial^2 f}{\\partial x_n^2}(a)\n\\end{bmatrix}.\n\\]\n\nThe Hessian is symmetric if \\(f\\) is twice differentiable.\n\n---\n\n## 12. **Implicit Function Theorem**\n\nLet \\(F(x, y) = 0\\) define an implicit function \\(y = g(x)\\). The theorem states that if \\(\\frac{\\partial F}{\\partial y}(x_0, y_0) \\neq 0\\) in a neighborhood of \\((x_0, y_0)\\), then there exists a function \\(g(x)\\) such that \\(F(x, g(x)) = 0\\) in that neighborhood.\n\n---\n\n## Summary\n\n| Concept                   | Single Variable                            | Multivariable                                        |\n|--------------------------|--------------------------------------------|------------------------------------------------------|\n| Limit                    | \\(\\lim_{x \\to a} f(x)\\)                    | \\(\\lim_{(x,y) \\to (a,b)} f(x,y)\\)                    |\n| Derivative                | \\(f'(x)\\), slope of tangent               | \\(\\frac{\\partial f}{\\partial x}\\), \\(\\frac{\\partial f}{\\partial y}\\) |\n| Directional Derivative     | N/A                                        | \\(D_{\\mathbf{u}} f(a) = \\nabla f(a) \\cdot \\mathbf{u}\\) |\n| Gradient                  | \\(f'(x)\\)                                  | \\(\\nabla f = \\left( \\frac{\\partial f}{\\partial x}, \\dots \\right)\\) |\n| Chain Rule                | Basic form                                | \\(\\frac{dz}{dt} = \\frac{\\partial f}{\\partial x} \\frac{dx}{dt} + \\frac{\\partial f}{\\partial y} \\frac{dy}{dt}\\) |\n| Multiple Integrals        | \\(\\int_a^b f(x)dx\\)                       | \\(\\iint_R f(x,y)dxdy = \\int_a^b \\int_{a(x,y)}^{b(x,y)} f \\, dy dx\\) |\n| Hessian                   | N/A                                        | \\(H_f = \\left( \\frac{\\partial^2 f}{\\partial x^2}, \\dots, \\frac{\\partial^2 f}{\\partial x_j \\partial x_k} \\right)\\) |\n\n---\n\nThis progression illustrates how the foundational ideas of calculus\u2014limits and derivatives\u2014extend naturally to higher dimensions, enriching our ability to analyze functions and solve problems in physics, engineering, economics, and more.\n\nIf you'd like to dive deeper into any specific topic (e.g., parametrized curves, vector fields, Green\u2019s theorem, or vector calculus), feel free to ask!"}
{"model": "mlx-community/LFM2-VL-3B-4bit", "status": "OK", "load_time_s": 1, "gen_time_s": 2.69, "prompt_tokens": 27, "completion_tokens": 597, "total_tokens": 624, "tokens_per_sec": 222.16, "content_preview": "Calculus is a branch of mathematics that deals with the study of continuous change. It comprises two major parts: differential calculus and integral calculus.   1. Limits:  In calculus, a limit is the value that a function approaches as the input approaches a certain point. It's a fundamental concep", "content": "Calculus is a branch of mathematics that deals with the study of continuous change. It comprises two major parts: differential calculus and integral calculus. \n\n1. Limits: \nIn calculus, a limit is the value that a function approaches as the input approaches a certain point. It's a fundamental concept in both differential and integral calculus.\n\nMathematically, we denote the limit of a function f(x) as x approaches 'a' by lim x->a f(x). This means we are looking for the value that f(x) approaches as x gets closer and closer to 'a'. \n\n2. Differential Calculus: \nDifferential calculus is concerned with the concept of the derivative of a function at a point. The derivative of a function f(x) at a point x=a (denoted as f'(a)) is defined as the limit of the difference quotient [f(x) - f(a)] / (x - a) as x approaches a.\n\nMathematically, this can be written as:\n\nf'(a) = lim(x->a) [f(x) - f(a)] / (x - a)\n\nThe derivative gives us the rate at which the function is changing at a given point, which is a crucial concept in physics, engineering, and many other fields.\n\n3. Integral Calculus: \nIntegral calculus is concerned with the concept of the integral of a function. The definite integral of f(x) from a to b, denoted as \u222b[a,b] f(x) dx, represents the area under the curve of f(x) between x=a and x=b.\n\nMathematically, it can be derived from the concept of the Riemann sum:\n\n\u222b[a,b] f(x) dx = lim(n->\u221e) \u03a3 [f(x_i) * \u0394x]\n\nwhere x_i are points in the partition of the interval [a,b] and \u0394x is the width of each partition.\n\n4. Multivariable Calculus:\nMultivariable calculus extends the concepts of differential and integral calculus to functions of several variables. The derivative of a function f(x, y) at a point (a, b) is a vector of partial derivatives.\n\nMathematically, it can be expressed as:\n\n\u2207f(a, b) = (\u2202f/\u2202x(a, b), \u2202f/\u2202y(a, b))\n\nwhere \u2202f/\u2202x and \u2202f/\u2202y are the partial derivatives of f with respect to x and y, respectively.\n\nThe definite integral in multivariable calculus is used to calculate volumes, surface areas, and other quantities in three-dimensional space. It can be written as:\n\n\u222b\u222b[D] f(x, y) dx dy\n\nwhere D is the region of integration.\n\nIn summary, calculus is a powerful mathematical tool that allows us to understand and describe continuous change. From limits to multivariable calculus, these concepts form the foundation of much of modern science and engineering."}
{"model": "mlx-community/lille-130m-instruct-8bit", "status": "OK", "load_time_s": 1, "gen_time_s": 2.05, "prompt_tokens": 27, "completion_tokens": 445, "total_tokens": 472, "tokens_per_sec": 216.8, "content_preview": "Calculus is a branch of mathematics that deals with the study of limits in a strict mathematical notation. It involves defining and describing functions as well as limits in order to get accurate and meaningful results. Here's how it works:  1. Estimation: This involves defining what a function is d", "content": "Calculus is a branch of mathematics that deals with the study of limits in a strict mathematical notation. It involves defining and describing functions as well as limits in order to get accurate and meaningful results. Here's how it works:\n\n1. Estimation: This involves defining what a function is defined as a function, which is a function that is defined as a function that can be defined as a function. In other words, as a function in terms of the meaning of its limits, also known as limits. This definition involves defining the properties of the limit, which is the function of limits. This definition of the function is often denoted by f(x) and is defined as the limit of the function f(x) as f(x) = f(x) for all possible values of x.\n\n2.\n\nThe functions in the limits of a function are defined as the limit of the function f(x) to the limit of the function f(x) where x is a real function. This definition of the function f(x) = x.\n\nThe limits of a function are defined as the limit of a limit, in the limit of a function:\n\n\na function, where x is defined as the limit of a function, where x is defined as the limit of a function, where x is defined as the limit of (f(x) as x approaches a limit.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n"}
{"model": "mlx-community/Ling-mini-2.0-4bit", "status": "OK", "load_time_s": 3, "gen_time_s": 15.18, "prompt_tokens": 27, "completion_tokens": 2354, "total_tokens": 2381, "tokens_per_sec": 155.09, "content_preview": "Certainly! Let's walk through **calculus concepts from limits through multivariable calculus**, using **rigorous mathematical notation**. We'll structure the material in a logical progression, starting from the foundational idea of a limit and moving up to multivariable and vector calculus.  ---  ##", "content": "Certainly! Let's walk through **calculus concepts from limits through multivariable calculus**, using **rigorous mathematical notation**. We'll structure the material in a logical progression, starting from the foundational idea of a limit and moving up to multivariable and vector calculus.\n\n---\n\n## 1. **Limits and Continuity**\n\n### 1.1. Single-Variable Limits\n\nWe begin with real-valued functions of a real variable.\n\nLet $ f: \\mathbb{R} \\to \\mathbb{R} $ be a function. The **limit** of $ f(x) $ as $ x \\to a $ is defined as:\n\n$$\n\\lim_{x \\to a} f(x) = L\n$$\n\nif for every $ \\varepsilon > 0 $, there exists a $ \\delta > 0 $ such that for all $ x $ satisfying $ 0< |x - a|< \\delta $, we have $ |f(x) - L|< \\varepsilon $.\n\nThis is the **\u03b5-\u03b4 definition** of a limit.\n\n---\n\n### 1.2. Continuity\n\nA function $ f $ is **continuous at $ a $** if:\n\n$$\n\\lim_{x \\to a} f(x) = f(a)\n$$\n\nEquivalently, for every $ \\varepsilon > 0 $, there exists $ \\delta > 0 $ such that if $ |x - a|< \\delta $, then $ |f(x) - f(a)|< \\varepsilon $.\n\n---\n\n### 1.3. Limits at Infinity\n\nWe also consider limits as $ x \\to \\infty $ or $ x \\to -\\infty $:\n\n$$\n\\lim_{x \\to \\infty} f(x) = L \\iff \\forall \\varepsilon > 0, \\exists M > 0 \\text{ such that } x > M \\Rightarrow |f(x) - L|< \\varepsilon\n$$\n\nSimilarly for $ x \\to -\\infty $.\n\n---\n\n## 2. **Differentiation**\n\n### 2.1. Derivatives\n\nThe derivative of $ f $ at $ a $ is defined as:\n\n$$\nf'(a) = \\lim_{h \\to 0} \\frac{f(a + h) - f(a)}{h}\n$$\n\nThis is the **slope of the tangent line** to the graph of $ f $ at $ x = a $.\n\n---\n\n### 2.2. Differentiability\n\nA function $ f $ is **differentiable at $ a $** if the above limit exists. Differentiability implies continuity.\n\n---\n\n### 2.3. Mean Value Theorem (MVT)\n\nIf $ f $ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $ c \\in (a, b) $ such that:\n\n$$\nf'(c) = \\frac{f(b) - f(a)}{b - a}\n$$\n\nThis is a fundamental result in differential calculus.\n\n---\n\n## 3. **Integrals**\n\n### 3.1. Antiderivatives and Indefinite Integrals\n\nA function $ F $ is an **antiderivative** of $ f $ on an interval $ I $ if $ F' = f $ on $ I $. The indefinite integral is:\n\n$$\n\\int f(x)\\,dx = F(x) + C\n$$\n\nwhere $ C $ is a constant.\n\n---\n\n### 3.2. Definite Integrals\n\nThe definite integral of $ f $ from $ a $ to $ b $ is defined as:\n\n$$\n\\int_a^b f(x)\\,dx = \\lim_{n \\to \\infty} \\sum_{i=1}^n f(x_i^*) \\Delta x\n$$\n\nwhere $ \\Delta x = \\frac{b - a}{n} $, and $ x_i^* \\in [x_{i-1}, x_i] $ (Riemann sum).\n\nIf the limit exists, $ f $ is **Riemann integrable** on $[a, b]$.\n\n---\n\n### 3.3. Fundamental Theorem of Calculus\n\nIf $ f $ is continuous on $[a, b]$, and $ F(x) = \\int_a^x f(t)\\,dt $, then $ F $ is differentiable and $ F'(x) = f(x) $.\n\nAlso, if $ F $ is an antiderivative of $ f $, then:\n\n$$\n\\int_a^b f(x)\\,dx = F(b) - F(a)\n$$\n\n---\n\n## 4. **Multivariable Calculus**\n\n### 4.1. Functions of Several Variables\n\nLet $ f: \\mathbb{R}^n \\to \\mathbb{R} $. We consider limits, continuity, and differentiability in higher dimensions.\n\n#### Partial Derivatives\n\nThe **partial derivative** of $ f $ with respect to $ x_i $ at point $ \\mathbf{a} = (a_1, \\dots, a_n) $ is:\n\n$$\n\\frac{\\partial f}{\\partial x_i}(\\mathbf{a}) = \\lim_{h \\to 0} \\frac{f(a_1, \\dots, a_i + h, \\dots, a_n) - f(\\mathbf{a})}{h}\n$$\n\n---\n\n### 4.2. Gradient Vector\n\nThe **gradient** of $ f $ is the vector of partial derivatives:\n\n$$\n\\nabla f(\\mathbf{a}) = \\left( \\frac{\\partial f}{\\partial x_1}(\\mathbf{a}), \\dots, \\frac{\\partial f}{\\partial x_n}(\\mathbf{a}) \\right)\n$$\n\nThe gradient points in the direction of maximum rate of increase of $ f $.\n\n---\n\n### 4.3. Directional Derivatives\n\nThe **directional derivative** of $ f $ at $ \\mathbf{a} $ in the direction of a unit vector $ \\mathbf{u} = (u_1, \\dots, u_n) $ is:\n\n$$\nD_{\\mathbf{u}}f(\\mathbf{a}) = \\nabla f(\\mathbf{a}) \\cdot \\mathbf{u} = \\sum_{i=1}^n \\frac{\\partial f}{\\partial x_i}(\\mathbf{a}) u_i\n$$\n\n---\n\n### 4.4. Tangent Planes and Linear Approximation\n\nIf $ f $ is differentiable at $ \\mathbf{a} $, then:\n\n$$\nf(\\mathbf{a} + \\mathbf{h}) \\approx f(\\mathbf{a}) + \\nabla f(\\mathbf{a}) \\cdot \\mathbf{h}\n$$\n\nThis is the **linear approximation** of $ f $ near $ \\mathbf{a} $.\n\n---\n\n### 4.5. Double Integrals and Change of Variables\n\nLet $ f: \\mathbb{R}^2 \\to \\mathbb{R} $. The double integral over a region $ D \\subset \\mathbb{R}^2 $ is:\n\n$$\n\\iint_D f(x, y)\\,dA\n$$\n\nUsing **Fubini\u2019s Theorem**, if $ f $ is continuous, then:\n\n$$\n\\iint_D f(x, y)\\,dA = \\int_{x \\in D} \\int_{y \\in D} f(x, y)\\,dy\\,dx\n$$\n\n---\n\n### 4.6. Change of Variables (Jacobian)\n\nFor a transformation $ \\mathbf{y} = \\mathbf{F}(\\mathbf{x}) $ from $ \\mathbb{R}^n $ to $ \\mathbb{R}^n $, the **Jacobian determinant** is:\n\n$$\nJ(\\mathbf{F}) = \\det\\left( \\frac{\\partial F_i}{\\partial x_j} \\right)\n$$\n\nThen, under a change of variables $ \\mathbf{x} = \\mathbf{G}(\\mathbf{y}) $, the integral becomes:\n\n$$\n\\iint_D f(x, y)\\,dA = \\iint_{D'} f(G_1(u,v), G_2(u,v)) \\cdot |J(u,v)| \\, du\\,dv\n$$\n\nThis is the essence of the **substitution theorem** in multiple integrals.\n\n---\n\n## 5. **Vector Calculus**\n\n### 5.1. Vector Fields and Scalar Fields\n\nA **scalar field** is a function $ f: \\mathbb{R}^n \\to \\mathbb{R} $, like temperature.\n\nA **vector field** is a function $ \\mathbf{F}: \\mathbb{R}^n \\to \\mathbb{R}^n $, like velocity field.\n\n---\n\n### 5.2. Line Integrals\n\nLet $ \\mathbf{F} = (P, Q) $ be a vector field in $ \\mathbb{R}^2 $, and $ C $ a curve parametrized by $ \\mathbf{r}(t) = (x(t), y(t)) $, $ a \\leq t \\leq b $. The **line integral** of $ \\mathbf{F} $ along $ C $ is:\n\n$$\n\\int_C \\mathbf{F} \\cdot d\\mathbf{r} = \\int_a^b \\left( P(x(t), y(t)) x'(t) + Q(x(t), y(t)) y'(t) \\right) dt\n$$\n\nThis represents the work done by $ \\mathbf{F} $ along $ C $.\n\n---\n\n### 5.3. Green\u2019s Theorem\n\nIf $ C $ is a positively oriented, piecewise-smooth, simple closed curve and $ D $ is the region bounded by $ C $, and $ \\mathbf{F} = (P, Q) $ has continuous partial derivatives on an open region containing $ D $, then:\n\n$$\n\\oint_C \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_D \\left( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right) dA\n$$\n\nGreen\u2019s Theorem links line integrals and double integrals.\n\n---\n\n### 5.4. Surface Integrals and Divergence Theorem\n\nLet $ \\mathbf{F} = (P, Q, R) $ be a vector field in $ \\mathbb{R}^3 $, and $ S $ be a surface oriented with unit normal $ \\mathbf{n} $. The **surface integral** of $ \\mathbf{F} $ over $ S $ is:\n\n$$\n\\iint_S \\mathbf{F} \\cdot d\\mathbf{S} = \\iint_S \\mathbf{F} \\cdot \\mathbf{n}\\, dS\n$$\n\nThe **Divergence Theorem** (Gauss's Theorem) states:\n\n$$\n\\iint_S \\mathbf{F} \\cdot \\mathbf{n}\\, dS = \\iiint_V \\nabla \\cdot \\mathbf{F}\\, dV\n$$\n\nwhere $ V $ is the volume bounded by $ S $, and $ \\nabla \\cdot \\mathbf{F} = \\frac{\\partial P}{\\partial x} + \\frac{\\partial Q}{\\partial y} + \\frac{\\partial R}{\\partial z} $.\n\n---\n\n### 5.5. Curl and Stokes\u2019 Theorem\n\nFor a vector field $ \\mathbf{F} = (P, Q, R) $ in $ \\mathbb{R}^3 $, the **curl** is:\n\n$$\n\\nabla \\times \\mathbf{F} = \\left( \\frac{\\partial R}{\\partial y} - \\frac{\\partial Q}{\\partial z}, \\frac{\\partial P}{\\partial z} - \\frac{\\partial R}{\\partial x}, \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right)\n$$\n\nStokes\u2019 Theorem relates the surface integral of the curl to the line integral around the boundary:\n\n$$\n\\iint_S (\\nabla \\times \\mathbf{F}) \\cdot \\mathbf{n}\\, dS = \\oint_{\\partial S} \\mathbf{F} \\cdot d\\mathbf{r}\n$$\n\n---\n\n## 6. **Advanced Topics**\n\n### 6.1. Optimization\n\nIn multivariable calculus, optimization involves finding extrema of functions $ f: \\mathbb{R}^n \\to \\mathbb{R} $.\n\n- **Critical points** occur where $ \\nabla f(\\mathbf{x}) = \\mathbf{0} $.\n- Use the **Hessian matrix** $ H $ to classify critical points:\n  - If $ H $ is positive definite, local minimum.\n  - If $ H $ is negative definite, local maximum.\n  - If indefinite, saddle point.\n\n---\n\n### 6.2. Parametric and Implicit Surfaces\n\n- A **parametric surface** is defined by $ \\mathbf{r}(u, v) = (x(u,v), y(u,v), z(u,v)) $.\n- An **implicit surface** is defined by $ F(x, y, z) = 0 $.\n\n---\n\n### 6.3. Differential Forms and Integration on Manifolds (Optional)\n\nIn advanced settings, calculus extends to integration over **manifolds** using **differential forms** and the **Stokes theorem** generalized:\n\n$$\n\\int_M d\\omega = \\int_{\\partial M} \\omega\n$$\n\nThis formalism underpins modern differential geometry and physics.\n\n---\n\n## Summary Table\n\n| Topic | Key Concept | Mathematical Notation |\n|-------|-------------|------------------------|\n| Limits | Definition of limit | $ \\lim_{x \\to a} f(x) = L $ |\n| Differentiability | Derivative definition | $ f'(a) = \\lim_{h \\to 0} \\frac{f(a+h)-f(a)}{h} $ |\n| Integrals | Riemann integral | $ \\int_a^b f(x)\\,dx $ |\n| Multivariable | Partial derivative | $ \\frac{\\partial f}{\\partial x} $ |\n| | Gradient | $ \\nabla f $ |\n| | Directional derivative | $ D_{\\mathbf{u}}f = \\nabla f \\cdot \\mathbf{u} $ |\n| | Line integral | $ \\int_C \\mathbf{F} \\cdot d\\mathbf{r} $ |\n| | Green\u2019s Theorem | $ \\oint_C \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_D (\\partial Q/\\partial x - \\partial P/\\partial y) dA $ |\n| | Divergence Theorem | $ \\iint_S \\mathbf{F} \\cdot \\mathbf{n} dS = \\iiint_V \\nabla \\cdot \\mathbf{F} dV $ |\n| | Stokes\u2019 Theorem | $ \\iint_S (\\nabla \\times \\mathbf{F}) \\cdot \\mathbf{n} dS = \\oint_{\\partial S} \\mathbf{F} \\cdot d\\mathbf{r} $ |\n\n---\n\nLet me know if you'd like a specific topic expanded further (e.g., proofs, examples, visualizations, or applications in physics/engineering)."}
{"model": "mlx-community/Llama-3.2-1B-Instruct-4bit", "status": "OK", "load_time_s": 2, "gen_time_s": 2.7, "prompt_tokens": 27, "completion_tokens": 731, "total_tokens": 758, "tokens_per_sec": 270.42, "content_preview": "**Limits** ================  Limits are a fundamental concept in multivariable calculus, which generalize the concept of limits in single-variable calculus. The **limit** of a function f(x, y, z,... ) at point (a, b, c,...) is defined as:   lim h \u2192 0 (f(a + h, b, c,..., d,...))) = (f(a, b, c,..., d,", "content": "**Limits**\n================\n\nLimits are a fundamental concept in multivariable calculus, which generalize the concept of limits in single-variable calculus. The **limit** of a function f(x, y, z,... ) at point (a, b, c,...) is defined as:\n\n lim h \u2192 0 (f(a + h, b, c,..., d,...))) = (f(a, b, c,..., d,...))) - (f(a, b, c,..., d,...))) / h\n\nIn other words, as the input (x, y, z,...) approaches the point (a, b, c,...), the function value approaches the limit of the function at the point (a, b, c,...).\n\n**Multivariable Limits**\n=====================\n\nMultivariable limits are similar to limits in single-variable calculus, but with the added complexity of multiple variables. The multivariable limit of a function f(x1, y1, z1,..., xn,...) at point (a1, b1,..., cn,...) is defined as:\n\n lim h1,..., hn \u2192 0 (f((a1 + h1, b1, c1,..., hn,...))) = (f(a1, b1, c1,..., hn,...))) - (f(a1, b1, c1,..., hn,...))) / h1 *... * h0\n\nNotice that the denominator h1... h0 is a product of multiple variables h1,..., hn.\n\n**Multivariable Derivatives**\n==========================\n\nMultivariable derivatives are a fundamental concept in multivariable calculus, which generalize the concept of derivatives in single-variable calculus. The multivariable derivative of a function f(x1, y1, z1,...) at point (a1, b1,..., cn,...) is defined as:\n\n lim h1,..., hn \u2192 0 (f((a1 + h1, b1, c1,..., hn,...))) / (h1 *... * hn)\n\nNotice that the denominator h1... hn is a product of multiple variables h1,..., hn.\n\n**Multivariable Integals**\n=====================\n\nMultivariable integrals are a fundamental concept in multivariable calculus, which generalize the concept of integrals in single-variable calculus. The multivariable integral of a function f(x1, y1, z1,...) at point (a1, b1,..., cn,...) is defined as:\n\n lim \u03c6 (f)((a1, b1, c1,..., \u03c6,...))) / \u03c6 = (f(a1, b1, c1,..., \u03c6,...))) - (f(a1, b1, c1,..., \u03c6,...))) / \u03c6\n\nNotice that the denominator \u03c6 is a constant.\n\n**Other Multivariable Concepts**\n================================\n\n*   **Multivariable Leibniz Rule**: The multivariable Leibniz rule is a fundamental concept in multivariable calculus, which relates the multivariable derivatives of a function. The Leibniz rule is defined as:\n\n \u2202f/\u2202(x1, y1, z1,..., xn) = (\u2202f/\u2202(x1, y1, z1,..., xn))(x1, y1, z1,..., xn) + (-1) \u2202^2f/\u2202(x1, y1, z1,..., xn)\u00b2\n\n\n*   **Multivariable Taylor Series**: The multivariable Taylor series is a fundamental concept in multivariable calculus, which relates the multivariable function at a point (a1, b1,..., cn,...) to the multivariable function at the point (a, b, c,..., d,...). The Taylor series is defined as:\n\n f(a1, b1,..., cn,...) = f(a, b, c,..., d,...) + (\u2202f/\u2202(x1, y1, z1,..., xn))(a, b, c,..., d) + (\u2202^2f/\u2202(x1, y1, z1,..., xn))(x1, y1, z1,..., xn)\u00b2 +... + (\u2202^(n-1) f/\u2202(x1, y1, z1,..., xn))(x1, y1, z1,..., xn)^(n-1)\n\n\nNote that the above notation is used to provide the solution to the problem asked by the user."}
{"model": "mlx-community/MiniMax-M2.5-5bit", "status": "OK", "load_time_s": 47, "gen_time_s": 63.03, "prompt_tokens": 27, "completion_tokens": 1991, "total_tokens": 2018, "tokens_per_sec": 31.59, "content_preview": "# Calculus: From Limits to Multivariable Calculus  ## 1. Limits and Continuity  ### 1.1 Limit of a Function For a function \\( f: \\mathbb{R} \\to \\mathbb{R} \\), the limit as \\( x \\) approaches \\( a \\) is \\( L \\) if: \\[ \\forall \\epsilon > 0, \\exists \\delta > 0 \\text{ such that } 0< |x - a|< \\delta \\imp", "content": "# Calculus: From Limits to Multivariable Calculus\n\n## 1. Limits and Continuity\n\n### 1.1 Limit of a Function\nFor a function \\( f: \\mathbb{R} \\to \\mathbb{R} \\), the limit as \\( x \\) approaches \\( a \\) is \\( L \\) if:\n\\[\n\\forall \\epsilon > 0, \\exists \\delta > 0 \\text{ such that } 0< |x - a|< \\delta \\implies |f(x) - L|< \\epsilon\n\\]\nDenoted: \\( \\lim_{x \\to a} f(x) = L \\)\n\n### 1.2 One-Sided Limits\n- Left-hand limit: \\( \\lim_{x \\to a^-} f(x) = L \\) means \\( \\forall \\epsilon > 0, \\exists \\delta > 0 \\) such that \\( a - \\delta< x< a \\implies |f(x) - L|< \\epsilon \\)\n- Right-hand limit: \\( \\lim_{x \\to a^+} f(x) = L \\) similarly defined for \\( a< x< a + \\delta \\)\n\n### 1.3 Continuity\nA function \\( f \\) is continuous at \\( a \\) if:\n\\[\n\\lim_{x \\to a} f(x) = f(a)\n\\]\nFormally: \\( \\forall \\epsilon > 0, \\exists \\delta > 0 \\) such that \\( |x - a|< \\delta \\implies |f(x) - f(a)|< \\epsilon \\)\n\n### 1.4 Intermediate Value Theorem\nIf \\( f \\) is continuous on \\([a, b]\\) and \\( N \\) is between \\( f(a) \\) and \\( f(b) \\), then \\( \\exists c \\in [a, b] \\) such that \\( f(c) = N \\).\n\n### 1.5 L'H\u00f4pital's Rule\nIf \\( \\lim_{x \\to a} \\frac{f(x)}{g(x)} \\) yields \\( \\frac{0}{0} \\) or \\( \\frac{\\infty}{\\infty} \\), then:\n\\[\n\\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\lim_{x \\to a} \\frac{f'(x)}{g'(x)}\n\\]\nprovided the limit on the right exists.\n\n---\n\n## 2. Differentiation\n\n### 2.1 Derivative Definition\nThe derivative of \\( f \\) at \\( a \\) is:\n\\[\nf'(a) = \\lim_{h \\to 0} \\frac{f(a + h) - f(a)}{h}\n\\]\nEquivalently:\n\\[\nf'(a) = \\lim_{x \\to a} \\frac{f(x) - f(a)}{x - a}\n\\]\n\n### 2.2 Differentiability and Continuity\nIf \\( f \\) is differentiable at \\( a \\), then \\( f \\) is continuous at \\( a \\). The converse is not necessarily true.\n\n### 2.3 Chain Rule\nIf \\( y = f(g(x)) \\), then:\n\\[\n\\frac{dy}{dx} = f'(g(x)) \\cdot g'(x)\n\\]\n\n### 2.4 Product Rule\n\\[\n(uv)' = u'v + uv'\n\\]\n\n### 2.5 Quotient Rule\n\\[\n\\left(\\frac{u}{v}\\right)' = \\frac{u'v - uv'}{v^2}\n\\]\n\n### 2.6 Mean Value Theorem\nIf \\( f \\) is continuous on \\([a, b]\\) and differentiable on \\( (a, b) \\), then \\( \\exists c \\in (a, b) \\) such that:\n\\[\nf'(c) = \\frac{f(b) - f(a)}{b - a}\n\\]\n\n### 2.7 Taylor Series\nIf \\( f \\) has derivatives of all orders at \\( a \\), then:\n\\[\nf(x) = \\sum_{n=0}^{\\infty} \\frac{f^{(n)}(a)}{n!}(x - a)^n\n\\]\nfor \\( x \\) in the radius of convergence.\n\n---\n\n## 3. Integration\n\n### 3.1 Riemann Integral\nThe Riemann integral of \\( f \\) on \\([a, b] \\) is:\n\\[\n\\int_a^b f(x) \\, dx = \\lim_{n \\to \\infty} \\sum_{i=1}^n f(x_i^*) \\Delta x_i\n\\]\nwhere \\( \\Delta x_i = \\frac{b-a}{n} \\) and \\( x_i^* \\in [x_{i-1}, x_i] \\) with \\( x_i = a + i\\Delta x \\).\n\n### 3.2 Fundamental Theorem of Calculus\n1. If \\( F(x) = \\int_a^x f(t) \\, dt \\), then \\( F'(x) = f(x) \\).\n2. If \\( F \\) is an antiderivative of \\( f \\), then:\n\\[\n\\int_a^b f(x) \\, dx = F(b) - F(a)\n\\]\n\n### 3.3 Integration Techniques\n- **Substitution**: \\( \\int f(g(x))g'(x) \\, dx = \\int f(u) \\, du \\) with \\( u = g(x) \\)\n- **Integration by parts**: \\( \\int u \\, dv = uv - \\int v \\, du \\)\n- **Partial fractions**: Decompose rational functions into simpler fractions\n\n### 3.4 Improper Integrals\n- Type I: Infinite limits\n  \\[\n  \\int_a^\\infty f(x) \\, dx = \\lim_{b \\to \\infty} \\int_a^b f(x) \\, dx\n  \\]\n- Type II: Infinite discontinuities\n  \\[\n  \\int_a^b f(x) \\, dx = \\lim_{c \\to b^-} \\int_a^c f(x) \\, dx\n  \\]\n\n---\n\n## 4. Sequences and Series\n\n### 4.1 Convergence of Sequences\nA sequence \\( \\{a_n\\} \\) converges to \\( L \\) if:\n\\[\n\\forall \\epsilon > 0, \\exists N \\in \\mathbb{N} \\text{ such that } n \\geq N \\implies |a_n - L|< \\epsilon\n\\]\nDenoted: \\( \\lim_{n \\to \\infty} a_n = L \\)\n\n### 4.2 Series Convergence\nThe series \\( \\sum_{n=1}^\\infty a_n \\) converges if the sequence of partial sums \\( S_N = \\sum_{n=1}^N a_n \\) converges.\n\n### 4.3 Convergence Tests\n- **Integral Test**: Compare with an integral\n- **Comparison Test**: Compare with a known series\n- **Ratio Test**: \\( \\lim_{n \\to \\infty} \\left| \\frac{a_{n+1}}{a_n} \\right| = L \\), converges if \\( L< 1 \\)\n- **Root Test**: \\( \\lim_{n \\to \\infty} \\sqrt[n]{|a_n|} = L \\), converges if \\( L< 1 \\)\n- **Alternating Series Test**: If \\( a_n \\) is decreasing and \\( \\lim_{n \\to \\infty} a_n = 0 \\), then \\( \\sum (-1)^n a_n \\) converges.\n\n### 4.4 Power Series\nA power series about \\( a \\):\n\\[\n\\sum_{n=0}^\\infty c_n (x - a)^n\n\\]\nhas a radius of convergence \\( R \\) such that it converges for \\( |x - a|< R \\) and diverges for \\( |x - a| > R \\).\n\n---\n\n## 5. Multivariable Calculus\n\n### 5.1 Functions of Several Variables\nA function \\( f: \\mathbb{R}^n \\to \\mathbb{R} \\) maps \\( \\mathbf{x} = (x_1, x_2, \\ldots, x_n) \\) to \\( f(\\mathbf{x}) \\).\n\n### 5.2 Limits in Several Variables\nThe limit of \\( f(\\mathbf{x}) \\) as \\( \\mathbf{x} \\to \\mathbf{a} \\) is \\( L \\) if:\n\\[\n\\forall \\epsilon > 0, \\exists \\delta > 0 \\text{ such that } 0< \\|\\mathbf{x} - \\mathbf{a}\\|< \\delta \\implies |f(\\mathbf{x}) - L|< \\epsilon\n\\]\n\n### 5.3 Partial Derivatives\nFor \\( f(x, y) \\), the partial derivative with respect to \\( x \\) is:\n\\[\nf_x = \\frac{\\partial f}{\\partial x} = \\lim_{h \\to 0} \\frac{f(x + h, y) - f(x, y)}{h}\n\\]\nSimilarly for \\( y \\).\n\n### 5.4 Gradient Vector\nFor \\( f: \\mathbb{R}^n \\to \\mathbb{R} \\), the gradient is:\n\\[\n\\nabla f = \\left( \\frac{\\partial f}{\\partial x_1}, \\frac{\\partial f}{\\partial x_2}, \\ldots, \\frac{\\partial f}{\\partial x_n} \\right)\n\\]\n\n### 5.5 Chain Rule\nIf \\( z = f(x, y) \\) and \\( x = g(t), y = h(t) \\), then:\n\\[\n\\frac{dz}{dt} = \\frac{\\partial f}{\\partial x} \\frac{dx}{dt} + \\frac{\\partial f}{\\partial y} \\frac{dy}{dt}\n\\]\n\n### 5.6 Directional Derivive\nThe directional derivative of \\( f \\) at \\( \\mathbf{a} \\) in direction \\( \\mathbf{u} \\) (unit vector) is:\n\\[\nD_{\\mathbf{u}} f(\\mathbf{a}) = \\nabla f(\\mathbf{a}) \\cdot \\mathbf{u}\n\\]\n\n### 5.7 Multiple Integrals\n\n#### Double Integral\n\\[\n\\iint_R f(x, y) \\, dA = \\lim_{m,n \\to \\infty} \\sum_{i=1}^m \\sum_{j=1}^n f(x_{ij}, y_{ij}) \\Delta A\n\\]\nwhere \\( \\Delta A = \\Delta x \\Delta y \\).\n\n#### Triple Integral\n\\[\n\\iiint_E f(x, y, z) \\, dV\n\\]\nwith \\( dV = dx \\, dy \\, dz \\) in Cartesian coordinates.\n\n#### Change of Variables\nFor transformations \\( (x, y) = (g(u, v), h(u, v) \\), the Jacobian is:\n\\[\nJ = \\frac{\\partial(x, y)}{\\partial(u, v)} = \\begin{vmatrix} x_u & x_v \\\\ y_u & y_v \\end{vmatrix}\n\\]\nThen:\n\\[\n\\iint_R f(x, y) \\, dA = \\iint_S f(g(u, v), h(u, v)) |J| \\, du \\, dv\n\\]\n\n### 5.8 Vector Calculus\n\n#### Line Integral\nFor a curve \\( C \\) parameterized by \\( \\mathbf{r}(t) \\), \\( a \\leq t \\leq b \\):\n- Scalar field: \\( \\int_C f \\, ds = \\int_a^b f(\\mathbf{r}(t)) \\|\\mathbf{r}'(t)\\| \\, dt \\)\n- Vector field: \\( \\int_C \\mathbf{F} \\cdot d\\mathbf{r} = \\int_a^b \\mathbf{F}(\\mathbf{r}(t)) \\cdot \\mathbf{r}'(t) \\, dt \\)\n\n#### Green's Theorem\nFor a positively oriented, piecewise-smooth simple closed curve \\( C \\) enclosing region \\( D \\):\n\\[\n\\oint_C (P \\, dx + Q \\, dy) = \\iint_D \\left( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right) \\, dA\n\\]\n\n#### Stokes' Theorem\nFor a surface \\( S \\) with boundary \\( \\partial S \\):\n\\[\n\\iint_S (\\nabla \\times \\mathbf{F}) \\cdot \\mathbf{n} \\, dS = \\oint_{\\partial S} \\mathbf{F} \\cdot d\\mathbf{r}\n\\]\n\n#### Divergence Theorem\nFor a solid \\( E \\) with boundary surface \\( S \\):\n\\[\n\\iiint_E (\\nabla \\cdot \\mathbf{F}) \\, dV = \\iint_S \\mathbf{F} \\cdot \\mathbf{n} \\, dS\n\\]\n\n---\n\nThis overview covers the fundamental concepts in calculus from single-variable to multivariable settings, with rigorous definitions and key theorems. Each section can be expanded with more examples and applications."}
{"model": "mlx-community/MiniMax-M2.5-6bit", "status": "OK", "load_time_s": 59, "gen_time_s": 98.39, "prompt_tokens": 27, "completion_tokens": 2832, "total_tokens": 2859, "tokens_per_sec": 28.78, "content_preview": "# Calculus: From Limits to Multivariable Calculus  ## 1. Limits and Continuity  ### 1.1 Limit of a Function  **Definition (\u03b5-\u03b4):**   Let \\( f: D \\subseteq \\mathbb{R} \\to \\mathbb{R} \\) and let \\( c \\) be a limit point of \\( D \\). We say   \\[ \\lim_{x \\to c} f(x) = L \\] if for every \\( \\varepsilon > 0 ", "content": "# Calculus: From Limits to Multivariable Calculus\n\n## 1. Limits and Continuity\n\n### 1.1 Limit of a Function\n\n**Definition (\u03b5-\u03b4):**  \nLet \\( f: D \\subseteq \\mathbb{R} \\to \\mathbb{R} \\) and let \\( c \\) be a limit point of \\( D \\). We say  \n\\[\n\\lim_{x \\to c} f(x) = L\n\\]\nif for every \\( \\varepsilon > 0 \\), there exists \\( \\delta > 0 \\) such that  \n\\[\n0< |x - c|< \\delta \\implies |f(x) - L|< \\varepsilon.\n\\]\n\n**One-sided limits:**  \n- Right-hand limit: \\( \\lim_{x \\to c^+} f(x) = L \\) if \\( \\forall \\varepsilon > 0, \\exists \\delta > 0: 0< x - c< \\delta \\implies |f(x) - L|< \\varepsilon \\).  \n- Left-hand limit: \\( \\lim_{x \\to c^-} f(x) = L \\) if \\( \\forall \\varepsilon > 0, \\exists \\delta > 0: 0< c - x< \\delta \\implies |f(x) - L|< \\varepsilon \\).\n\n**Theorem:** \\( \\lim_{x \\to c} f(x) = L \\) exists if and only if \\( \\lim_{x \\to c^+} f(x) = \\lim_{x \\to c^-} f(x) = L \\).\n\n### 1.2 Properties of Limits\n\nIf \\( \\lim_{x \\to c} f(x) = L \\) and \\( \\lim_{x \\to c} g(x) = M \\), then:\n\n1. \\( \\lim_{x \\to c} [f(x) \\pm g(x)] = L \\pm M \\).\n2. \\( \\lim_{x \\to c} [f(x) \\cdot g(x)] = L \\cdot M \\).\n3. \\( \\lim_{x \\to c} \\frac{f(x)}{g(x)} = \\frac{L}{M} \\) provided \\( M \\neq 0 \\).\n4. \\( \\lim_{x \\to c} f(x)^n = L^n \\) for \\( n \\in \\mathbb{N} \\).\n5. \\( \\lim_{x \\to c} \\sqrt[n]{f(x)} = \\sqrt[n]{L} \\) if \\( L \\geq 0 \\) for even \\( n \\).\n\n### 1.3 Continuity\n\n**Definition:** A function \\( f \\) is continuous at \\( c \\) if  \n\\[\n\\lim_{x \\to c} f(x) = f(c).\n\\]\n\nEquivalently: \\( \\forall \\varepsilon > 0, \\exists \\delta > 0 \\) such that \\( |x - c|< \\delta \\implies |f(x) - f(c)|< \\varepsilon \\).\n\n**Theorem (Intermediate Value):** If \\( f \\) is continuous on \\( [a, b] \\) and \\( N \\) lies between \\( f(a) \\) and \\( f(b) \\), then there exists \\( c \\in (a, b) \\) such that \\( f(c) = N \\).\n\n---\n\n## 2. Differentiation\n\n### 2.1 Derivative\n\n**Definition:** The derivative of \\( f \\) at \\( c \\) is  \n\\[\nf'(c) = \\lim_{h \\to 0} \\frac{f(c + h) - f(c)}{h},\n\\]\nprovided the limit exists. If the limit exists, \\( f \\) is differentiable at \\( c \\).\n\n**Alternative form:**  \n\\[\nf'(c) = \\lim_{x \\to c} \\frac{f(x) - f(c)}{x - c}.\n\\]\n\n**Geometric interpretation:** \\( f'(c) \\) is the slope of the tangent line at \\( (c, f(c)) \\).\n\n### 2.2 Rules of Differentiation\n\n1. **Constant multiple:** \\( (cf)' = cf' \\).\n2. **Sum/difference:** \\( (f \\pm g)' = f' \\pm g' \\).\n3. **Product:** \\( (fg)' = f'g + fg' \\).\n4. **Quotient:** \\( \\left( \\frac{f}{g} \\right)' = \\frac{f'g - fg'}{g^2} \\), for \\( g \\neq 0 \\).\n5. **Chain rule:** If \\( y = f(g(x)) \\), then \\( \\frac{dy}{dx} = f'(g(x)) \\cdot g'(x) \\).\n\n### 2.3 Mean Value Theorem\n\n**Theorem:** If \\( f \\) is continuous on \\( [a, b] \\) and differentiable on \\( (a, b) \\), then there exists \\( c \\in (a, b) \\) such that  \n\\[\nf'(c) = \\frac{f(b) - f(a)}{b - a}.\n\\]\n\n### 2.4 Higher-Order Derivatives\n\nThe \\( n \\)-th derivative:  \n\\[\nf^{(n)}(x) = \\frac{d^n}{dx^n} f(x).\n\\]\n\n### 2.5 Taylor Series\n\nIf \\( f \\) has derivatives of all orders at \\( c \\), then  \n\\[\nf(x) = \\sum_{n=0}^{\\infty} \\frac{f^{(n)}(c)}{n!} (x - c)^n,\n\\]\nprovided the series converges.\n\n---\n\n## 3. Integration\n\n### 3.1 Antiderivative and Indefinite Integral\n\n**Definition:** A function \\( F \\) is an antiderivative of \\( f \\) if \\( F' = f \\). We write  \n\\[\n\\int f(x) \\, dx = F(x) + C,\n\\]\nwhere \\( C \\) is the constant of integration.\n\n### 3.2 Definite Integral\n\n**Definition (Riemann):** For \\( f \\) bounded on \\( [a, b] \\), let \\( P = \\{a = x_0< x_1< \\cdots< x_n = b\\} \\) be a partition, and \\( \\xi_i \\in [x_{i-1}, x_i] \\). The Riemann sum is  \n\\[\nS(P, f) = \\sum_{i=1}^{n} f(\\xi_i) \\Delta x_i, \\quad \\Delta x_i = x_i - x_{i-1}.\n\\]\nThe definite integral is  \n\\[\n\\int_a^b f(x) \\, dx = \\lim_{\\|P\\| \\to 0} \\sum_{i=1}^{n} f(\\xi_i) \\Delta x_i,\n\\]\nprovided the limit exists (where \\( \\|P\\| = \\max \\Delta x_i \\)).\n\n### 3.3 Fundamental Theorem of Calculus\n\n1. If \\( F \\) is an antiderivative of \\( f \\) on \\( [a, b] \\), then  \n   \\[\n   \\int_a^b f(x) \\, dx = F(b) - F(a).\n   \\]\n\n2. If \\( f \\) is continuous on \\( [a, b] \\), then the function  \n   \\[\n   g(x) = \\int_a^x f(t) \\, dt\n   \\]\n   is differentiable, and \\( g'(x) = f(x) \\).\n\n### 3.4 Techniques of Integration\n\n1. **Substitution:** Let \\( u = g(x) \\), then \\( du = g'(x) dx \\).  \n   \\[\n   \\int f(g(x)) g'(x) \\, dx = \\int f(u) \\, du.\n   \\]\n\n2. **Integration by parts:**  \n   \\[\n   \\int u \\, dv = uv - \\int v \\, du.\n   \\]\n\n3. **Partial fractions:** For rational functions \\( \\frac{P(x)}{Q(x)} \\), decompose into simpler fractions.\n\n### 3.5 Improper Integrals\n\n**Type I (unbounded interval):**  \n\\[\n\\int_a^{\\infty} f(x) \\, dx = \\lim_{b \\to \\infty} \\int_a^b f(x) \\, dx.\n\\]\n\n**Type II (unbounded integrand):**  \nIf \\( f \\) has a singularity at \\( c \\in [a, b] \\),  \n\\[\n\\int_a^b f(x) \\, dx = \\lim_{t \\to c^-} \\int_a^t f(x) \\, dx + \\lim_{t \\to c^+} \\int_t^b f(x) \\, dx.\n\\]\n\n---\n\n## 4. Sequences and Series\n\n### 4.1 Sequences\n\n**Definition:** A sequence \\( \\{a_n\\} \\) converges to \\( L \\) if  \n\\[\n\\forall \\varepsilon > 0, \\exists N \\in \\mathbb{N} \\text{ such that } n \\geq N \\implies |a_n - L|< \\varepsilon.\n\\]\nWe write \\( \\lim_{n \\to \\infty} a_n = L \\).\n\n### 4.2 Series\n\nConsider \\( \\sum_{n=1}^{\\infty} a_n \\). The partial sums are \\( S_N = \\sum_{n=1}^{N} a_n \\). The series converges if \\( \\lim_{N \\to \\infty} S_N \\) exists.\n\n**Tests for convergence:**\n\n1. **Geometric series:** \\( \\sum_{n=0}^{\\infty} ar^n \\) converges if \\( |r|< 1 \\), with sum \\( \\frac{a}{1-r} \\).\n\n2. **Integral test:** If \\( f \\) is positive, decreasing, and continuous, then \\( \\sum f(n) \\) converges if and only if \\( \\int_1^{\\infty} f(x) \\, dx \\) converges.\n\n3. **Comparison test:** If \\( 0 \\leq a_n \\leq b_n \\) and \\( \\sum b_n \\) converges, then \\( \\sum a_n \\) converges.\n\n4. **Ratio test:** Let \\( L = \\lim_{n \\to \\infty} \\left| \\frac{a_{n+1}}{a_n} \\right| \\). Then the series converges if \\( L< 1 \\), diverges if \\( L > 1 \\), and is inconclusive if \\( L = 1 \\).\n\n5. **Root test:** Let \\( L = \\lim_{n \\to \\infty} \\sqrt[n]{|a_n|} \\). Then the series converges if \\( L< 1 \\), diverges if \\( L > 1 \\), and is inconclusive if \\( L = 1 \\).\n\n### 4.3 Power Series\n\nA power series about \\( c \\) is  \n\\[\n\\sum_{n=0}^{\\infty} a_n (x - c)^n.\n\\]\nThe radius of convergence \\( R \\) is given by  \n\\[\nR = \\frac{1}{\\limsup_{n \\to \\infty} \\sqrt[n]{|a_n|}}.\n\\]\nWithin \\( |x - c|< R \\), the series converges absolutely.\n\n---\n\n## 5. Multivariable Calculus\n\n### 5.1 Functions of Several Variables\n\nLet \\( f: D \\subseteq \\mathbb{R}^n \\to \\mathbb{R} \\). For \\( \\mathbf{x} = (x_1, \\ldots, x_n) \\), we write \\( f(\\mathbf{x}) \\).\n\n### 5.2 Limits and Continuity\n\n**Limit:** \\( \\lim_{\\mathbf{x} \\to \\mathbf{a}} f(\\mathbf{x}) = L \\) if  \n\\[\n\\forall \\varepsilon > 0, \\exists \\delta > 0 \\text{ such that } 0< \\|\\mathbf{x} - \\mathbf{a}\\|< \\delta \\implies |f(\\mathbf{x}) - L|< \\varepsilon.\n\\]\n\n**Continuity:** \\( f \\) is continuous at \\( \\mathbf{a} \\) if \\( \\lim_{\\mathbf{x} \\to \\mathbf{a}} f(\\mathbf{x}) = f(\\mathbf{a}) \\).\n\n### 5.3 Partial Derivatives\n\nFor \\( f(x, y) \\), the partial derivative with respect to \\( x \\) is  \n\\[\nf_x = \\frac{\\partial f}{\\partial x} = \\lim_{h \\to 0} \\frac{f(x + h, y) - f(x, y)}{h}.\n\\]\nSimilarly, \\( f_y = \\frac{\\partial f}{\\partial y} \\).\n\n**Higher-order partial derivatives:**  \n\\( f_{xx} = \\frac{\\partial^2 f}{\\partial x^2} \\), \\( f_{xy} = \\frac{\\partial^2 f}{\\partial y \\partial x} \\), etc.\n\n** Clairaut's theorem:** If \\( f_{xy} \\) and \\( f_{yx} \\) are continuous, then \\( f_{xy} = f_{yx} \\).\n\n### 5.4 Gradient and Directional Derivative\n\nFor \\( f: \\mathbb{R}^n \\to \\mathbb{R} \\), the gradient is  \n\\[\n\\nabla f = \\left( \\frac{\\partial f}{\\partial x_1}, \\ldots, \\frac{\\partial f}{\\partial x_n} \\right).\n\\]\n\nThe directional derivative in the direction of a unit vector \\( \\mathbf{u} \\) is  \n\\[\nD_{\\mathbf{u}} f(\\mathbf{a}) = \\nabla f(\\mathbf{a}) \\cdot \\mathbf{u}.\n\\]\n\n### 5.5 Multiple Integrals\n\n**Double integral:**  \n\\[\n\\iint_R f(x, y) \\, dA = \\iint_R f(x, y) \\, dx \\, dy.\n\\]\n\n**Triple integral:**  \n\\[\n\\iiint_E f(x, y, z) \\, dV.\n\\]\n\n**Change of variables:** For transformations \\( (u, v) \\to (x, y) \\) with Jacobian \\( J = \\frac{\\partial(x, y)}{\\partial(u, v)} \\),  \n\\[\n\\iint_R f(x, y) \\, dx \\, dy = \\iint_S f(x(u, v), y(u, v)) |J| \\, du \\, dv.\n\\]\n\n### 5.6 Vector Calculus\n\n#### 5.6.1 Vector Fields\n\nA vector field \\( \\mathbf{F} = (P, Q, R) \\) in \\( \\mathbb{R}^3 \\) is a function \\( \\mathbf{F}: \\mathbb{R}^3 \\to \\mathbb{R}^3 \\).\n\n#### 5.6.2 Gradient, Divergence, Curl\n\n- **Gradient:** \\( \\nabla f = \\left( \\frac{\\partial f}{\\partial x}, \\frac{\\partial f}{\\partial y}, \\frac{\\partial f}{\\partial z} \\right) \\).\n- **Divergence:** \\( \\nabla \\cdot \\mathbf{F} = \\frac{\\partial P}{\\partial x} + \\frac{\\partial Q}{\\partial y} + \\frac{\\partial R}{\\partial z} \\).\n- **Curl:** \\( \\nabla \\times \\mathbf{F} = \\left( \\frac{\\partial R}{\\partial y} - \\frac{\\partial Q}{\\partial z}, \\frac{\\partial P}{\\partial z} - \\frac{\\partial R}{\\partial x}, \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right) \\).\n\n#### 5.6.3 Line Integrals\n\nFor a curve \\( C \\) parameterized by \\( \\mathbf{r}(t) \\), \\( a \\leq t \\leq b \\),  \n\\[\n\\int_C \\mathbf{F} \\cdot d\\mathbf{r} = \\int_a^b \\mathbf{F}(\\mathbf{r}(t)) \\cdot \\mathbf{r}'(t) \\, dt.\n\\]\n\n#### 5.6.4 Surface Integrals\n\nFor a surface \\( S \\) parameterized by \\( \\mathbf{r}(u, v) \\),  \n\\[\n\\iint_S f \\, dS = \\iint_D f(\\mathbf{r}(u, v)) \\|\\mathbf{r}_u \\times \\mathbf{r}_v\\| \\, du \\, dv.\n\\]\n\nFor flux:  \n\\[\n\\iint_S \\mathbf{F} \\cdot \\mathbf{n} \\, dS = \\iint_D \\mathbf{F}(\\mathbf{r}(u, v)) \\cdot (\\mathbf{r}_u \\times \\mathbf{r}_v) \\, du \\, dv.\n\\]\n\n### 5.7 Fundamental Theorems\n\n#### 5.7.1 Green's Theorem\n\nFor a positively oriented, piecewise-smooth curve \\( C \\) enclosing region \\( D \\),  \n\\[\n\\oint_C (P \\, dx + Q \\, dy) = \\iint_D \\left( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right) \\, dA.\n\\]\n\n#### 5.7.2 Stokes' Theorem\n\nFor a surface \\( S \\) with boundary curve \\( C \\),  \n\\[\n\\int_C \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_S (\\nabla \\times \\mathbf{F}) \\cdot \\mathbf{n} \\, dS.\n\\]\n\n#### 5.7.3 Divergence Theorem\n\nFor a solid \\( E \\) with boundary surface \\( S \\),  \n\\[\n\\iint_S \\mathbf{F} \\cdot \\mathbf{n} \\, dS = \\iiint_E (\\nabla \\cdot \\mathbf{F}) \\, dV.\n\\]\n\n---\n\nThis covers the essential topics from single-variable calculus to multivariable calculus, emphasizing rigorous mathematical notation and key theorems."}
{"model": "mlx-community/NVIDIA-Nemotron-3-Nano-30B-A3B-4bit", "status": "OK", "load_time_s": 6, "gen_time_s": 51.5, "prompt_tokens": 27, "completion_tokens": 2703, "total_tokens": 2730, "tokens_per_sec": 52.49, "content_preview": "Below is a compact, yet fully rigorous, roadmap of calculus from the **\u03b5\u2013\u03b4 definition of a limit** through the core ideas of **single\u2011variable differentiation and integration**, and finally to the machinery of **multivariable calculus**.   All statements are written with the standard set\u2011theoretic n", "content": "Below is a compact, yet fully rigorous, roadmap of calculus from the **\u03b5\u2013\u03b4 definition of a limit** through the core ideas of **single\u2011variable differentiation and integration**, and finally to the machinery of **multivariable calculus**.  \nAll statements are written with the standard set\u2011theoretic notation; where a proof is sketched, the essential logical steps are indicated.\n\n---\n\n## 1. Limits and Continuity in \\(\\mathbb{R}\\)\n\n### 1.1 Definition (\u03b5\u2013\u03b4 limit of a real function)\n\nLet \\(f:\\mathbb{R}\\to\\mathbb{R}\\) and let \\(a\\in\\mathbb{R}\\).  \n\\[\n\\lim_{x\\to a}f(x)=L\\quad\\Longleftrightarrow\\quad\n\\forall\\varepsilon>0\\;\\exists\\delta>0\\; \\text{s.t.}\\;0<|x-a|<\\delta \\implies |f(x)-L|<\\varepsilon .\n\\]\n\n### 1.2 Continuity\n\n\\(f\\) is **continuous at \\(a\\)** iff \\(\\displaystyle \\lim_{x\\to a}f(x)=f(a)\\).  \nEquivalently, for every \\(\\varepsilon>0\\) there exists \\(\\delta>0\\) such that  \n\\( |x-a|<\\delta \\implies |f(x)-f(a)|<\\varepsilon\\).\n\n### 1.3 Sequential characterisation\n\n\\(f\\) is continuous at \\(a\\) iff for every sequence \\((x_n)\\) with \\(x_n\\to a\\) (and \\(x_n\\neq a\\) eventually) we have \\(f(x_n)\\to f(a)\\).\n\n---\n\n## 2. Differentiation in One Variable\n\n### 2.1 Derivative as a limit\n\nFor \\(f:\\mathbb{R}\\to\\mathbb{R}\\) and \\(a\\in\\mathbb{R}\\),\n\\[\nf'(a):=\\lim_{h\\to0}\\frac{f(a+h)-f(a)}{h},\n\\]\nprovided the limit exists (as a finite real number).\n\n### 2.2 Existence \u21d2 Continuity\n\nIf \\(f\\) is differentiable at \\(a\\) then \\(f\\) is continuous at \\(a\\).  \n*Proof sketch*: Write \\(f(a+h)=f(a)+h\\frac{f(a+h)-f(a)}{h}\\).  \nGiven \\(\\varepsilon>0\\), choose \\(\\delta\\) for the derivative limit with \\(\\varepsilon/2\\) and use the triangle inequality.\n\n### 2.3 Fundamental derivative rules (formal statements)\n\n| Rule | Formal statement |\n|------|------------------|\n| **Linearity** | \\((c_1f+c_2g)'(a)=c_1f'(a)+c_2g'(a)\\). |\n| **Product** | \\((fg)'(a)=f'(a)g(a)+f(a)g'(a)\\). |\n| **Quotient** | \\(\\displaystyle\\Big(\\frac{f}{g}\\Big)'(a)=\\frac{f'(a)g(a)-f(a)g'(a)}{[g(a)]^{2}}\\) provided \\(g(a)\\neq0\\). |\n| **Chain rule** | \\((f\\circ g)'(a)=f'(g(a))\\cdot g'(a)\\). |\n| **Power rule** | \\(\\displaystyle \\frac{d}{dx}x^{n}=nx^{\\,n-1}\\) for any integer \\(n\\). |\n\nAll of the above follow directly from the definition of the derivative and algebraic manipulation of limits.\n\n---\n\n## 3. Integration in One Variable\n\n### 3.1 Riemann integral\n\nLet \\(f:[a,b]\\to\\mathbb{R}\\) be bounded. For a partition \\(P=\\{x_0,\\dots ,x_n\\}\\) with mesh \\(\\|P\\|=\\max_i (x_i-x_{i-1})\\), choose sample points \\(\\xi_i\\in[x_{i-1},x_i]\\). The **Riemann sum** is  \n\\[\nS(P,\\xi)=\\sum_{i=1}^{n} f(\\xi_i)\\,\\Delta x_i,\\qquad \\Delta x_i=x_i-x_{i-1}.\n\\]\n\\(f\\) is **Riemann integrable** on \\([a,b]\\) if there exists \\(L\\in\\mathbb{R}\\) such that for every \\(\\varepsilon>0\\) there is \\(\\delta>0\\) with \\(\\|P\\|<\\delta\\) implying \\(|S(P,\\xi)-L|<\\varepsilon\\).  \nWe denote \\(\\displaystyle \\int_a^b f(x)\\,dx = L\\).\n\n### 3.2 Fundamental Theorem of Calculus (FTC)\n\n1. **First part**: If \\(F\\) is an antiderivative of \\(f\\) on \\([a,b]\\) (i.e. \\(F' = f\\)), then  \n   \\[\n   \\int_a^b f(x)\\,dx = F(b)-F(a).\n   \\]\n\n2. **Second part**: Define \\(F(x)=\\int_a^x f(t)\\,dt\\). Then \\(F\\) is continuous on \\([a,b]\\), differentiable on \\((a,b)\\) with \\(F'(x)=f(x)\\).  \n\nBoth statements rely on the completeness of \\(\\mathbb{R}\\) and the definition of the Riemann integral.\n\n---\n\n## 4. Sequences and Series of Functions\n\n### 4.1 Pointwise convergence\n\nA sequence of functions \\((f_n)\\) on a set \\(E\\) **converges pointwise** to \\(f\\) if for each \\(x\\in E\\),\n\\[\n\\forall\\varepsilon>0\\;\\exists N\\; \\text{s.t.}\\; n\\ge N \\implies |f_n(x)-f(x)|<\\varepsilon .\n\\]\n\n### 4.2 Uniform convergence\n\n\\((f_n)\\) converges **uniformly** to \\(f\\) on \\(E\\) iff  \n\\[\n\\forall\\varepsilon>0\\;\\exists N\\; \\text{s.t.}\\; n\\ge N \\implies \\sup_{x\\in E}|f_n(x)-f(x)|<\\varepsilon .\n\\]\n\nUniform convergence preserves continuity, integrability, and differentiability under standard hypotheses.\n\n### 4.3 Power series (radius of convergence)\n\nFor a power series \\(\\displaystyle \\sum_{k=0}^\\infty a_k (x-c)^k\\) define its **radius of convergence**\n\\[\nR = \\frac{1}{\\displaystyle \\limsup_{k\\to\\infty} \\sqrt[k]{|a_k|}} \\in [0,\\infty].\n\\]\nInside \\((-c+R,c-R)\\) the series converges absolutely; at the boundary one must examine convergence case\u2011by\u2011case.\n\n---\n\n## 5. Multivariable Calculus\n\nAll definitions below are for maps defined on open subsets of \\(\\mathbb{R}^n\\).\n\n### 5.1 Partial derivatives\n\nLet \\(f:\\mathbb{R}^n\\to\\mathbb{R}\\). For each coordinate \\(i\\in\\{1,\\dots,n\\}\\) and point \\(a\\in\\mathbb{R}^n\\),\n\\[\n\\frac{\\partial f}{\\partial x_i}(a) := \\lim_{h\\to0}\\frac{f(a+he_i)-f(a)}{h},\n\\]\nwhere \\(e_i\\) is the \\(i\\)-th standard basis vector.\n\nIf all first\u2011order partials exist in a neighbourhood of \\(a\\) and are continuous there, then \\(f\\) is **\\(C^1\\)** at \\(a\\).\n\n### 5.2 Differential (total derivative)\n\nA function \\(f:\\mathbb{R}^n\\to\\mathbb{R}^m\\) is **differentiable** at \\(a\\) if there exists a linear map \\(L:\\mathbb{R}^n\\to\\mathbb{R}^m\\) such that  \n\\[\n\\lim_{h\\to0}\\frac{\\|f(a+h)-f(a)-L(h)\\|}{\\|h\\|}=0 .\n\\]\nThe linear map \\(L\\) is denoted \\(Df(a)\\) or \\(df_a\\). When \\(m=1\\) this linear map can be identified with the gradient vector \\(\\nabla f(a)\\in\\mathbb{R}^n\\) via\n\\[\nDf(a)(h)=\\nabla f(a)\\cdot h.\n\\]\n\n### 5.3 Jacobian matrix and determinant\n\nFor \\(F:\\mathbb{R}^n\\to\\mathbb{R}^m\\) with component functions \\(F_i\\), the **Jacobian matrix** at \\(a\\) is the \\(m\\times n\\) matrix\n\\[\nJ_F(a)=\\begin{pmatrix}\n\\displaystyle\\frac{\\partial F_1}{\\partial x_1}(a) & \\cdots & \\displaystyle\\frac{\\partial F_1}{\\partial x_n}(a)\\\\\n\\vdots & & \\vdots\\\\\n\\displaystyle\\frac{\\partial F_m}{\\partial x_1}(a) & \\cdots & \\displaystyle\\frac{\\partial F_m}{\\partial x_n}(a)\n\\end{pmatrix}.\n\\]\nIf \\(n=m\\) and \\(\\det J_F(a)\\neq0\\), the **Inverse Function Theorem** guarantees a local inverse that is also differentiable.\n\n### 5.4 Multivariable Taylor\u2019s theorem\n\nLet \\(f:\\mathbb{R}^n\\to\\mathbb{R}\\) have continuous partial derivatives up to order \\(k\\) on a convex neighbourhood of \\(a\\). Then for each \\(h\\in\\mathbb{R}^n\\)\n\\[\nf(a+h)=\\sum_{|\\alpha|\\le k}\\frac{1}{\\alpha!}\\, D^\\alpha f(a)\\,h^\\alpha + R_{k+1}(h),\n\\]\nwhere \\(\\alpha=(\\alpha_1,\\dots,\\alpha_n)\\) is a multi\u2011index, \\(|\\alpha|=\\alpha_1+\\dots+\\alpha_n\\), \\(D^\\alpha\\) denotes the corresponding partial derivative, and the remainder satisfies\n\\[\n\\|R_{k+1}(h)\\|\\le C\\|h\\|^{k+1}\n\\]\nfor some constant \\(C\\) depending on the \\((k+1)\\)-st derivatives on the line segment \\([a,a+h]\\).\n\n### 5.5 Gradient, divergence, and Laplacian\n\nFor a vector field \\(\\mathbf{F}:\\mathbb{R}^n\\to\\mathbb{R}^n\\):\n\\[\n\\operatorname{div}\\mathbf{F} = \\sum_{i=1}^{n}\\frac{\\partial F_i}{\\partial x_i},\n\\qquad\n\\operatorname{curl}\\mathbf{F} \\text{ (in } \\mathbb{R}^3\\text{)} = \\nabla\\times\\mathbf{F}.\n\\]\nThe **Laplacian** of a scalar field \\(f\\) is \\(\\Delta f = \\operatorname{div}(\\nabla f)\\).\n\n### 5.6 Line and surface integrals\n\n* **Line integral of a scalar field**:  \n  \\[\n  \\int_{\\gamma} f\\,ds = \\int_a^b f(\\gamma(t))\\,\\|\\gamma'(t)\\|\\,dt,\n  \\]\n  where \\(\\gamma:[a,b]\\to\\mathbb{R}^n\\) is a smooth curve.\n\n* **Line integral of a vector field** (work):  \n  \\[\n  \\int_{\\gamma}\\mathbf{F}\\cdot d\\mathbf{r}= \\int_a^b \\mathbf{F}(\\gamma(t))\\cdot\\gamma'(t)\\,dt.\n  \\]\n\n* **Surface integral** (for a parametrized surface \\(\\mathbf{r}(u,v)\\)):\n  \\[\n  \\iint_S \\mathbf{F}\\cdot d\\mathbf{S}= \\iint_D \\mathbf{F}(\\mathbf{r}(u,v))\\cdot\\big(\\mathbf{r}_u\\times\\mathbf{r}_v\\big)\\,du\\,dv.\n  \\]\n\n### 5.7 Classical integral theorems\n\n| Theorem | Statement (in \\(\\mathbb{R}^3\\)) |\n|---------|----------------------------------|\n| **Fundamental Theorem of Calculus (gradient form)** | \\(\\displaystyle \\int_{\\gamma} \\nabla f\\cdot d\\mathbf{r}=f(\\mathbf{r}(b))-f(\\mathbf{r}(a)).\\) |\n| **Green\u2019s Theorem** | \\(\\displaystyle \\oint_{C} (P\\,dx+Q\\,dy)=\\iint_{D}\\Big(\\frac{\\partial Q}{\\partial x}-\\frac{\\partial P}{\\partial y}\\Big)\\,dA.\\) |\n| **Stokes\u2019 Theorem** | \\(\\displaystyle \\oint_{\\partial S}\\mathbf{F}\\cdot d\\mathbf{r}= \\iint_{S} (\\nabla\\times\\mathbf{F})\\cdot \\mathbf{n}\\,dS.\\) |\n| **Divergence (Gauss) Theorem** | \\(\\displaystyle \\iiint_{V} \\nabla\\cdot\\mathbf{F}\\,dV = \\iint_{\\partial V}\\mathbf{F}\\cdot\\mathbf{n}\\,dS.\\) |\n\nThese hold under the usual smoothness hypotheses (continuous partial derivatives on the relevant domains).\n\n---\n\n## 6. Quick Reference of Notation\n\n| Symbol | Meaning |\n|--------|----------|\n| \\(\\lim_{x\\to a}\\) | Limit as \\(x\\) approaches \\(a\\) (approach from any direction). |\n| \\(\\epsilon,\\delta\\) | Positive tolerances used in the definition of limits. |\n| \\(f'(a)\\) | Derivative of \\(f\\) at \\(a\\). |\n| \\(\\nabla f\\) | Gradient vector of a scalar field \\(f\\). |\n| \\(Df(a)\\) or \\(df_a\\) | Differential (total derivative) of \\(f\\) at \\(a\\). |\n| \\(J_F(a)\\) | Jacobian matrix of \\(F\\) at \\(a\\). |\n| \\(\\displaystyle\\int_a^b\\) | Riemann integral over \\([a,b]\\). |\n| \\(\\sum_{i=1}^n\\) | Summation over \\(n\\) indices. |\n| \\(\\alpha!\\) | Factorial of a multi\u2011index \\(\\alpha\\) defined multiplicatively. |\n| \\(\\|\\cdot\\|\\) | Norm in \\(\\mathbb{R}^n\\) (usually Euclidean). |\n| \\(\\mathbf{e}_i\\) | Standard basis vector in \\(\\mathbb{R}^n\\). |\n| \\(R\\) | Radius of convergence of a power series. |\n\n---\n\n### Closing Remark\n\nThis overview connects the **\u03b5\u2013\u03b4 foundations** of limits with the **differential and integral calculus** in one and several variables, emphasizing the precise logical structure behind each operation. The same principles extend to more abstract settings (Banach spaces, manifolds, distributions), but the formulas and theorems above capture the essential rigor used throughout classical and modern analysis."}
{"model": "mlx-community/Qwen2.5-0.5B-Instruct-4bit", "status": "OK", "load_time_s": 1, "gen_time_s": 13.91, "prompt_tokens": 27, "completion_tokens": 3114, "total_tokens": 3141, "tokens_per_sec": 223.84, "content_preview": "Calculus is the study of quantities or changes in quantities that can be described in terms of derivatives and integrals. Let\u2019s dive into limits through multivariable calculus with rigorous mathematical notation.  ### 1. Definition of a limit A limit is the value that a function approaches as the va", "content": "Calculus is the study of quantities or changes in quantities that can be described in terms of derivatives and integrals. Let\u2019s dive into limits through multivariable calculus with rigorous mathematical notation.\n\n### 1. Definition of a limit\nA limit is the value that a function approaches as the variable approaches a fixed point, or as the input approaches a limit.\n\n### 2. Definitions of right hand limit and left hand limit\nA limit is defined as a value that the function approaches as the variable approaches a point, or as the input approaches a limit.\n\n### 3. Right hand limit: \n- For a function \\(f(x)\\), the right hand limit as \\(x\\) approaches \\(a\\) is defined as:\n  \\[\n  \\lim_{x \\to a^+} f(x) = \\text{ the value that the function approaches as } x \\to a\n  \\]\n- Here, \\(a\\) is a fixed point in the domain of the function.\n\n### 4. Left hand limit: \n- For a function \\(f(x)\\), the left hand limit as \\(x\\) approaches \\(a\\) is defined as:\n  \\[\n  \\lim_{x \\to a^-} f(x) = \\text{ the value that the function approaches as } x \\to a\n  \\]\n- Here, \\(a\\) is a fixed point in the domain of the function.\n\n### 5. Definition of a limit point\nA limit point is a point in the domain of a function such that the function approaches a value as the variable approaches that point.\n\n### 6. Examples:\n- **Example 1**: \n  \\[\n  \\lim_{x \\to 3} \\frac{x^2 - 9}{x - 3} = \\lim_{x \\to 3} \\frac{(x - 3)(x + 3)}{x - 3} = x + 3 \\quad \\text{as} \\quad x \\to 3\n  \\]\n- **Example 2**:\n  \\[\n  \\lim_{x \\to 1} \\frac{x - 1}{x - 1} = \\lim_{x \\to 1} 1 \\quad \\text{as} \\quad x \\to 1\n  \\]\n\n### 7. Theorem on the limit of a function\nIf we have a function \\(f(x)\\), for any real number \\(L\\) such that \\(f(a) = L\\), then\n\\[\n\\lim_{x \\to a} f(x) = L\n\\]\n\n### 8. Theorem on the limit of a sequence\nIf we have a sequence \\(\\{x_n\\}\\) such that \\(\\lim_{n \\to \\infty} x_n = L\\), then\n\\[\n\\lim_{n \\to \\infty} f(x_n) = L\n\\]\n\n### 9. Theorem on the limit of a series\nIf we have a series \\(\\sum_{k=1}^\\infty a_k\\) such that \\(\\lim_{n \\to \\infty} a_n = L\\), then\n\\[\n\\lim_{n \\to \\infty} \\frac{a_n}{n} = 0\n\\]\n\n### 10. Theorem on the limit of a number\nIf we have a number \\(L\\) such that \\(L \\leq M\\), then\n\\[\n\\lim_{x \\to \\infty} f(x) = \\infty \\text{ or } \\infty \\text{ or } -\\infty \\text{ or } \\infty \\text{ or } \\infty\n\\]\n\n### 11. Theorem on the limit of a set\nIf we have a set \\(\\{x_n\\}\\) such that \\(\\lim_{n \\to \\infty} x_n = L\\), then\n\\[\n\\lim_{n \\to \\infty} x_n = L\n\\]\n\n### 12. Theorems on the limit of a function of a variable\nIf we have a function \\(f(x)\\) such that \\(\\lim_{x \\to a} f(x) = L\\), then\n\\[\n\\lim_{x \\to a} f(x) = L\n\\]\n\n### 13. Theorem on the limit of a series of values\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = 0\n\\]\n\n### 14. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 15. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = -\\infty\n\\]\n\n### 16. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 17. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 18. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 19. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 20. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 21. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 22. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 23. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 24. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 25. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 26. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 27. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 28. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 29. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 30. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 31. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 32. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 33. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 34. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 35. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 36. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 37. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 38. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 39. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 40. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 41. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 42. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 43. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 44. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 45. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 46. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 47. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 48. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 49. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 50. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\nTheorem on the limit of a number is a fundamental theorem in calculus that can be used to determine the limit of a number. Theorem on the limit of a set is a general theorem that can be used to determine the limit of a set.\n\n### 51. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 52. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 53. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 54. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 55. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 56. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 57. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 58. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 59. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 60. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 61. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 62. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 63. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 64. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 65. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 66. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 67. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 68. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 69. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 70. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 71. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 72. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 73. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 74. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 75. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 76. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 77. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 78. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 79. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 80. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 81. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 82. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 83. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 84. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 85. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 86. Theorem on the limit of a set\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\n### 87. Theorem on the limit of a number\n\\[\n\\lim_{x \\to \\infty} \\frac{a_1 + a_2 + \\ldots + a_n}{n} = \\infty\n\\]\n\nThese theorems on the limit of a number are fundamental because they provide a way to determine the limits of numbers, which in turn can be used to determine the limits of functions and series.\n\n### Conclusion\n\nCalculus is a powerful tool for solving problems in many areas of mathematics and its applications. Theorems on the limit of"}
{"model": "mlx-community/Qwen3-0.6B-4bit", "status": "OK", "load_time_s": 2, "gen_time_s": 8.52, "prompt_tokens": 27, "completion_tokens": 2878, "total_tokens": 2905, "tokens_per_sec": 337.8, "content_preview": "To explore calculus concepts from limits through multivariable calculus, let's break down the key ideas and connect them through examples and mathematical notation:  ### 1. **Limits and the Concept of Partial Derivatives**    - **Limit as x approaches 0 of f(x, 0)**:        This is equivalent to the", "content": "To explore calculus concepts from limits through multivariable calculus, let's break down the key ideas and connect them through examples and mathematical notation:\n\n### 1. **Limits and the Concept of Partial Derivatives**\n   - **Limit as x approaches 0 of f(x, 0)**:  \n     This is equivalent to the value of the function at the point (0, 0) when variables approach each other. In multivariable calculus, this is closely tied to partial derivatives.  \n     For example, if we define a function $ f(x, y) $, then the partial derivative $ \\frac{\\partial f}{\\partial x} $ at $ (0, 0) $ represents the limit as $ x $ approaches 0 of $ f(x, 0) $. Similarly, the partial derivative $ \\frac{\\partial f}{\\partial y} $ at $ (0, 0) $ represents the limit as $ y $ approaches 0 of $ f(0, y) $.\n\n   - **Partial Derivatives and Standard Derivatives**:  \n     In multivariable calculus, the partial derivatives of a function $ f(x, y) $, such as $ \\frac{\\partial f}{\\partial x} $, represent the rate of change of the function with respect to each variable. These derivatives are essentially the limits taken as variables approach each other in the domain of the function.\n\n### 2. **Example: The Function $ f(x, y) = x^2 + y^2 $**\n   - **Partial Derivatives**:\n     - $ \\frac{\\partial f}{\\partial x} $ at $ (0, 0) $: This is the derivative of $ f $ with respect to $ x $, which is $ 2x $. As $ x \\to 0 $, this derivative approaches 0, matching the limit as $ x \\to 0 $ of $ f(x, 0) $.\n     - $ \\frac{\\partial f}{\\partial y} $ at $ (0, 0) $: Here, $ 2y $, and as $ y \\to 0 $, this derivative also approaches 0, matching the limit as $ y \\to 0 $ of $ f(0, y) $.\n\n   - **Multivariable Limit**:  \n     In this example, the function $ f(x, y) = x^2 + y^2 $ has its domain restricted to the domain $ x \\in \\mathbb{R}, y \\in \\mathbb{R} $, and the limit as $ x $ and $ y $ approach each other (e.g., $ x, y \\to 0 $) is simply $ f(0, 0) = 0 $.\n\n### 3. **Connection Through Partial Derivatives**\n   - The partial derivatives of a function in two variables, such as $ f(x, y) $, allow us to express the relationship between variables as limits. This is often used in optimization problems or in the study of functions in multiple dimensions.\n\n### 4. **Generalization to Multivariable Functions**\n   - When considering functions in higher dimensions, the concept of limits becomes more general. For example, in three dimensions, the limit as $ x \\to 0 $ and $ y \\to 0 $ of $ f(x, y) $ in a 3D space would be related to the partial derivatives of the function evaluated at those points.\n\n### Summary and Conclusion\nIn multivariable calculus, the relationship between limits and partial derivatives is foundational. The partial derivatives of a function in multiple dimensions, when evaluated at points where variables approach each other (e.g., $ x \\to 0 $, $ y \\to 0 $), represent the limits that are closely tied to the function's behavior in those dimensions. This relationship is consistent and visually evident through examples like the function $ f(x, y) = x^2 + y^2 $, where the partial derivatives at the origin (0, 0) give the limits of the function at those points.\n\nThis setup highlights how limits in multivariable calculus, through the lens of partial derivatives, help us understand the relationships between variables and their respective limits in a multidimensional space."}
{"model": "mlx-community/Qwen3-30B-A3B-4bit", "status": "OK", "load_time_s": 6, "gen_time_s": 37.31, "prompt_tokens": 27, "completion_tokens": 3030, "total_tokens": 3057, "tokens_per_sec": 81.22, "content_preview": "### **1. Limits and Continuity**  #### **Definition of a Limit (Single Variable)** Let $ f: \\mathbb{R} \\to \\mathbb{R} $, and let $ a \\in \\mathbb{R} $. We say: $$ \\lim_{x \\to a} f(x) = L $$ if for every $ \\varepsilon > 0 $, there exists $ \\delta > 0 $ such that for all $ x \\in \\mathbb{R} $, $ 0< |x -", "content": "### **1. Limits and Continuity**\n\n#### **Definition of a Limit (Single Variable)**\nLet $ f: \\mathbb{R} \\to \\mathbb{R} $, and let $ a \\in \\mathbb{R} $. We say:\n$$\n\\lim_{x \\to a} f(x) = L\n$$\nif for every $ \\varepsilon > 0 $, there exists $ \\delta > 0 $ such that for all $ x \\in \\mathbb{R} $, $ 0< |x - a|< \\delta $ implies $ |f(x) - L|< \\varepsilon $.\n\n#### **Continuity at a Point**\nA function $ f $ is **continuous at $ a $** if:\n$$\n\\lim_{x \\to a} f(x) = f(a).\n$$\nEquivalently, for all $ \\varepsilon > 0 $, there exists $ \\delta > 0 $ such that $ |x - a|< \\delta $ implies $ |f(x) - f(a)|< \\varepsilon $.\n\n#### **Limit Laws**\nFor $ \\lim_{x \\to a} f(x) = L $ and $ \\lim_{x \\to a} g(x) = M $, we have:\n- $ \\lim_{x \\to a} [f(x) + g(x)] = L + M $\n- $ \\lim_{x \\to a} [f(x) \\cdot g(x)] = L \\cdot M $\n- $ \\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\frac{L}{M} $ (if $ M \\ne 0 $).\n\n---\n\n### **2. Derivatives**\n\n#### **Definition of the Derivative**\nThe derivative of $ f $ at $ a $ is:\n$$\nf'(a) = \\lim_{h \\to 0} \\frac{f(a + h) - f(a)}{h},\n$$\nprovided the limit exists. This represents the instantaneous rate of change of $ f $ at $ a $.\n\n#### **Differentiability Implies Continuity**\nIf $ f $ is differentiable at $ a $, then $ f $ is continuous at $ a $.\n\n#### **Rules of Differentiation**\n- **Chain Rule**: If $ f $ is differentiable at $ a $ and $ g $ is differentiable at $ f(a) $, then:\n  $$\n  (g \\circ f)'(a) = g'(f(a)) \\cdot f'(a).\n  $$\n- **Product Rule**: $ (fg)' = f'g + fg' $.\n- **Quotient Rule**: $ \\left(\\frac{f}{g}\\right)' = \\frac{f'g - fg'}{g^2} $ (if $ g \\ne 0 $).\n\n#### **Higher-Order Derivatives**\nThe $ n $-th derivative of $ f $ is denoted $ f^{(n)} $, where:\n$$\nf^{(n)}(x) = \\frac{d^n}{dx^n} f(x).\n$$\n\n---\n\n### **3. Integrals**\n\n#### **Riemann Integral (Single Variable)**\nLet $ f $ be defined on $ [a, b] $. The Riemann integral is:\n$$\n\\int_a^b f(x) \\, dx = \\lim_{\\|P\\| \\to 0} \\sum_{i=1}^n f(x_i^*) \\Delta x_i,\n$$\nwhere $ P $ is a partition of $ [a, b] $, $ \\Delta x_i = x_i - x_{i-1} $, and $ x_i^* \\in [x_{i-1}, x_i] $.\n\n#### **Fundamental Theorem of Calculus**\nIf $ f $ is continuous on $ [a, b] $, and $ F $ is an antiderivative of $ f $ (i.e., $ F' = f $), then:\n$$\n\\int_a^b f(x) \\, dx = F(b) - F(a).\n$$\nAdditionally, for $ F(x) = \\int_a^x f(t) \\, dt $, we have $ F'(x) = f(x) $.\n\n#### **Improper Integrals**\nFor $ \\int_a^\\infty f(x) \\, dx $, define:\n$$\n\\int_a^\\infty f(x) \\, dx = \\lim_{b \\to \\infty} \\int_a^b f(x) \\, dx,\n$$\nprovided the limit exists.\n\n---\n\n### **4. Multivariable Calculus**\n\n#### **Limits in $ \\mathbb{R}^n $**\nLet $ f: \\mathbb{R}^n \\to \\mathbb{R} $. We say:\n$$\n\\lim_{\\mathbf{x} \\to \\mathbf{a}} f(\\mathbf{x}) = L\n$$\nif for every $ \\varepsilon > 0 $, there exists $ \\delta > 0 $ such that $ \\|\\mathbf{x} - \\mathbf{a}\\|< \\delta $ implies $ |f(\\mathbf{x}) - L|< \\varepsilon $.\n\n#### **Partial Derivatives**\nFor $ f: \\mathbb{R}^n \\to \\mathbb{R} $, the **partial derivative** of $ f $ with respect to $ x_i $ at $ \\mathbf{a} $ is:\n$$\n\\frac{\\partial f}{\\partial x_i}(\\mathbf{a}) = \\lim_{h \\to 0} \\frac{f(\\mathbf{a} + h \\mathbf{e}_i) - f(\\mathbf{a})}{h},\n$$\nwhere $ \\mathbf{e}_i $ is the standard basis vector.\n\n#### **Gradient**\nThe **gradient** of $ f $ at $ \\mathbf{a} $ is:\n$$\n\\nabla f(\\mathbf{a}) = \\left( \\frac{\\partial f}{\\partial x_1}(\\mathbf{a}), \\dots, \\frac{\\partial f}{\\partial x_n}(\\mathbf{a}) \\right).\n$$\n\n#### **Directional Derivative**\nThe **directional derivative** of $ f $ at $ \\mathbf{a} $ in the direction of a unit vector $ \\mathbf{u} $ is:\n$$\nD_{\\mathbf{u}} f(\\mathbf{a}) = \\nabla f(\\mathbf{a}) \\cdot \\mathbf{u}.\n$$\n\n#### **Total Derivative**\nA function $ f: \\mathbb{R}^n \\to \\mathbb{R}^m $ is **differentiable at $ \\mathbf{a} $** if there exists a linear transformation $ Df(\\mathbf{a}) $ (represented by the Jacobian matrix) such that:\n$$\n\\lim_{\\mathbf{h} \\to 0} \\frac{\\|f(\\mathbf{a} + \\mathbf{h}) - f(\\mathbf{a}) - Df(\\mathbf{a})\\mathbf{h}\\|}{\\|\\mathbf{h}\\|} = 0.\n$$\n\n#### **Chain Rule in Multivariable**\nIf $ f: \\mathbb{R}^n \\to \\mathbb{R}^m $ and $ g: \\mathbb{R}^m \\to \\mathbb{R}^p $, then the derivative of $ g \\circ f $ at $ \\mathbf{a} $ is:\n$$\nD(g \\circ f)(\\mathbf{a}) = Dg(f(\\mathbf{a})) \\cdot Df(\\mathbf{a}).\n$$\n\n---\n\n### **5. Multiple Integrals**\n\n#### **Double and Triple Integrals**\nFor $ f: \\mathbb{R}^2 \\to \\mathbb{R} $, the double integral over a region $ D \\subset \\mathbb{R}^2 $ is:\n$$\n\\iint_D f(x, y) \\, dA = \\lim_{\\text{partitions} \\to 0} \\sum_{i,j} f(x_{ij}^*, y_{ij}^*) \\Delta A_{ij}.\n$$\nSimilarly for triple integrals over $ \\mathbb{R}^3 $.\n\n#### **Fubini's Theorem**\nIf $ f $ is continuous on $ [a, b] \\times [c, d] $, then:\n$$\n\\iint_{[a,b] \\times [c,d]} f(x, y) \\, dx \\, dy = \\int_a^b \\left( \\int_c^d f(x, y) \\, dy \\right) dx.\n$$\n\n#### **Change of Variables**\nFor a transformation $ \\mathbf{T}: \\mathbb{R}^n \\to \\mathbb{R}^n $, the integral becomes:\n$$\n\\iiint_{D} f(\\mathbf{x}) \\, d\\mathbf{x} = \\iiint_{D'} f(\\mathbf{T}(\\mathbf{u})) \\cdot |\\det D\\mathbf{T}(\\mathbf{u})| \\, d\\mathbf{u}.\n$$\n\n---\n\n### **6. Vector Calculus Theorems**\n\n#### **Green's Theorem**\nFor a positively oriented, piecewise-smooth closed curve $ C $ and region $ D \\subset \\mathbb{R}^2 $:\n$$\n\\oint_C (P \\, dx + Q \\, dy) = \\iint_D \\left( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right) dx \\, dy.\n$$\n\n#### **Stokes' Theorem**\nFor a surface $ S \\subset \\mathbb{R}^3 $ with boundary $ \\partial S $:\n$$\n\\int_S \\nabla \\times \\mathbf{F} \\cdot d\\mathbf{S} = \\oint_{\\partial S} \\mathbf{F} \\cdot d\\mathbf{r}.\n$$\n\n#### **Divergence Theorem (Gauss's Theorem)**\nFor a volume $ V \\subset \\mathbb{R}^3 $ with boundary $ \\partial V $:\n$$\n\\iiint_V \\nabla \\cdot \\mathbf{F} \\, dV = \\iint_{\\partial V} \\mathbf{F} \\cdot d\\mathbf{S}.\n$$\n\n---\n\n### **7. Key Concepts and Theorems**\n\n- **Mean Value Theorem**: If $ f $ is continuous on $ [a, b] $ and differentiable on $ (a, b) $, then there exists $ c \\in (a, b) $ such that:\n  $$\n  f'(c) = \\frac{f(b) - f(a)}{b - a}.\n  $$\n- **Taylor Series**: For $ f $ infinitely differentiable at $ a $,\n  $$\n  f(x) = \\sum_{n=0}^\\infty \\frac{f^{(n)}(a)}{n!} (x - a)^n.\n  $$\n- **Implicit Function Theorem**: If $ F(x, y) = 0 $ and $ \\frac{\\partial F}{\\partial y} \\ne 0 $, then $ y $ can be expressed as a function of $ x $ near $ (a, b) $.\n\n---\n\nThis framework provides a rigorous foundation for calculus, emphasizing formal definitions, theorems, and notation. Each concept builds on the previous, ensuring consistency across single-variable and multivariable settings."}
{"model": "mlx-community/Qwen3-Coder-Next-4bit", "status": "OK", "load_time_s": 16, "gen_time_s": 76.06, "prompt_tokens": 27, "completion_tokens": 3088, "total_tokens": 3115, "tokens_per_sec": 40.6, "content_preview": "# A Rigorous Introduction to Calculus: From Limits to Multivariable Calculus  ## 1. Limits  ### 1.1 \u03b5-\u03b4 Definition of a Limit  Let $f: D \\subseteq \\mathbb{R} \\to \\mathbb{R}$ and let $a$ be a limit point of $D$. We say that the limit of $f(x)$ as $x$ approaches $a$ is $L$, written: $$\\lim_{x \\to a} f", "content": "# A Rigorous Introduction to Calculus: From Limits to Multivariable Calculus\n\n## 1. Limits\n\n### 1.1 \u03b5-\u03b4 Definition of a Limit\n\nLet $f: D \\subseteq \\mathbb{R} \\to \\mathbb{R}$ and let $a$ be a limit point of $D$. We say that the limit of $f(x)$ as $x$ approaches $a$ is $L$, written:\n$$\\lim_{x \\to a} f(x) = L$$\nif for every $\\varepsilon > 0$, there exists a $\\delta > 0$ such that whenever $x \\in D$ and $0< |x - a|< \\delta$, then $|f(x) - L|< \\varepsilon$.\n\n### 1.2 One-Sided Limits\n\nThe right-hand limit of $f(x)$ as $x$ approaches $a$ is $L$, written:\n$$\\lim_{x \\to a^+} f(x) = L$$\nif for every $\\varepsilon > 0$, there exists $\\delta > 0$ such that whenever $a< x< a + \\delta$, then $|f(x) - L|< \\varepsilon$.\n\nSimilarly, the left-hand limit $\\lim_{x \\to a^-} f(x) = L$ if for every $\\varepsilon > 0$, there exists $\\delta > 0$ such that whenever $a - \\delta< x< a$, then $|f(x) - L|< \\varepsilon$.\n\n### 1.3 Infinite Limits\n\nWe say $\\lim_{x \\to a} f(x) = \\infty$ if for every $M > 0$, there exists $\\delta > 0$ such that $0< |x - a|< \\delta$ implies $f(x) > M$.\n\nSimilarly, $\\lim_{x \\to \\infty} f(x) = L$ if for every $\\varepsilon > 0$, there exists $N > 0$ such that $x > N$ implies $|f(x) - L|< \\varepsilon$.\n\n## 2. Continuity\n\nA function $f: D \\subseteq \\mathbb{R} \\to \\mathbb{R}$ is continuous at $a \\in D$ if:\n$$\\lim_{x \\to a} f(x) = f(a)$$\n\nEquivalently, $f$ is continuous at $a$ if for every $\\varepsilon > 0$, there exists $\\delta > 0$ such that $|x - a|< \\delta$ and $x \\in D$ implies $|f(x) - f(a)|< \\varepsilon$.\n\nA function is continuous on $D$ if it is continuous at every point $a \\in D$.\n\n**Theorem (Algebra of Continuous Functions):** If $f$ and $g$ are continuous at $a$, then:\n- $f + g$ is continuous at $a$\n- $f \\cdot g$ is continuous at $a$\n- $f/g$ is continuous at $a$ provided $g(a) \\neq 0$\n- If $h$ is continuous at $f(a)$, then $h \\circ f$ is continuous at $a$\n\n## 3. Differentiation in One Variable\n\n### 3.1 Definition of the Derivative\n\nLet $f: D \\subseteq \\mathbb{R} \\to \\mathbb{R}$ and let $a$ be an interior point of $D$. The derivative of $f$ at $a$ is defined as:\n$$f'(a) = \\lim_{h \\to 0} \\frac{f(a+h) - f(a)}{h}$$\nif this limit exists.\n\nEquivalently:\n$$f'(a) = \\lim_{x \\to a} \\frac{f(x) - f(a)}{x - a}$$\n\n### 3.2 Differentiability Implies Continuity\n\n**Theorem:** If $f$ is differentiable at $a$, then $f$ is continuous at $a$.\n\n*Proof:* If $f$ is differentiable at $a$, then:\n$$\\lim_{x \\to a} [f(x) - f(a)] = \\lim_{x \\to a} \\frac{f(x) - f(a)}{x - a} \\cdot (x - a) = f'(a) \\cdot 0 = 0$$\nso $\\lim_{x \\to a} f(x) = f(a)$, and $f$ is continuous at $a$.\n\n### 3.3 Differentiation Rules\n\nLet $f, g$ be differentiable at $x$, and let $c \\in \\mathbb{R}$:\n\n- **Constant Rule:** $(c)' = 0$\n- **Power Rule:** $(x^n)' = nx^{n-1}$ for $n \\in \\mathbb{N}$\n- **Sum Rule:** $(f + g)' = f' + g'$\n- **Product Rule:** $(fg)' = f'g + fg'$\n- **Quotient Rule:** $(f/g)' = \\frac{f'g - fg'}{g^2}$ (where $g \\neq 0$)\n- **Chain Rule:** If $h(x) = f(g(x))$, then $h'(x) = f'(g(x))g'(x)$\n\n### 3.4 Mean Value Theorem\n\n**Theorem (Mean Value Theorem):** If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $c \\in (a, b)$ such that:\n$$f'(c) = \\frac{f(b) - f(a)}{b - a}$$\n\n### 3.5 Taylor's Theorem\n\n**Theorem (Taylor's Theorem with Lagrange Remainder):** If $f$ is $n+1$ times differentiable on an interval containing $a$, then for each $x$ in the interval, there exists $c$ between $a$ and $x$ such that:\n$$f(x) = \\sum_{k=0}^{n} \\frac{f^{(k)}(a)}{k!} (x - a)^k + R_n(x)$$\nwhere $R_n(x) = \\frac{f^{(n+1)}(c)}{(n+1)!} (x - a)^{n+1}$\n\n## 4. Integration in One Variable\n\n### 4.1 Riemann Integral\n\nLet $f: [a, b] \\to \\mathbb{R}$ be bounded. A partition $P$ of $[a, b]$ is a finite set $\\{x_0, x_1, \\dots, x_n\\}$ where $a = x_0< x_1< \\dots< x_n = b$.\n\nFor each subinterval $[x_{i-1}, x_i]$, let $m_i = \\inf\\{f(x) : x \\in [x_{i-1}, x_i]\\}$ and $M_i = \\sup\\{f(x) : x \\in [x_{i-1}, x_i]\\}$.\n\nThe lower sum is $L(P, f) = \\sum_{i=1}^n m_i (x_i - x_{i-1})$  \nThe upper sum is $U(P, f) = \\sum_{i=1}^n M_i (x_i - x_{i-1})$\n\n**Definition:** $f$ is Riemann integrable on $[a, b]$ if:\n$$\\sup_P L(P, f) = \\inf_P U(P, f)$$\nIn this case, the common value is denoted $\\int_a^b f(x)\\,dx$.\n\n### 4.2 Fundamental Theorem of Calculus\n\n**Theorem (FTC Part I):** If $f$ is Riemann integrable on $[a, b]$ and $F(x) = \\int_a^x f(t)\\,dt$, then $F$ is continuous on $[a, b]$. If $f$ is continuous at $c \\in [a, b]$, then $F$ is differentiable at $c$ and $F'(c) = f(c)$.\n\n**Theorem (FTC Part II):** If $f$ is continuous on $[a, b]$ and $F$ is an antiderivative of $f$ on $[a, b]$, then:\n$$\\int_a^b f(x)\\,dx = F(b) - F(a)$$\n\n## 5. Sequences and Series\n\n### 5.1 Convergence of Sequences\n\nA sequence $(a_n)$ converges to $L$ if for every $\\varepsilon > 0$, there exists $N \\in \\mathbb{N}$ such that $n > N$ implies $|a_n - L|< \\varepsilon$.\n\n**Theorem (Cauchy Criterion):** A sequence $(a_n)$ converges if and only if for every $\\varepsilon > 0$, there exists $N \\in \\mathbb{N}$ such that $m, n > N$ implies $|a_m - a_n|< \\varepsilon$.\n\n### 5.2 Convergence of Series\n\nThe series $\\sum_{n=1}^{\\infty} a_n$ converges if the sequence of partial sums $s_N = \\sum_{n=1}^{N} a_n$ converges.\n\n**Comparison Test:** If $0 \\leq a_n \\leq b_n$ for all $n$ and $\\sum b_n$ converges, then $\\sum a_n$ converges.\n\n**Ratio Test:** If $\\limsup_{n \\to \\infty} \\left|\\frac{a_{n+1}}{a_n}\\right| = L$, then:\n- If $L< 1$, $\\sum a_n$ converges absolutely\n- If $L > 1$, $\\sum a_n$ diverges\n- If $L = 1$, the test is inconclusive\n\n**Root Test:** If $\\limsup_{n \\to \\infty} \\sqrt[n]{|a_n|} = L$, then the same conclusions as the ratio test hold.\n\n**Alternating Series Test:** If $a_n \\geq 0$, $a_{n+1} \\leq a_n$, and $\\lim_{n \\to \\infty} a_n = 0$, then $\\sum (-1)^n a_n$ converges.\n\n## 6. Vector Spaces and Multivariable Functions\n\n### 6.1 Euclidean Space\n\nLet $\\mathbb{R}^n$ denote the set of all $n$-tuples of real numbers. For $\\mathbf{x} = (x_1, x_2, \\dots, x_n)$ and $\\mathbf{y} = (y_1, y_2, \\dots, y_n)$ in $\\mathbb{R}^n$:\n\n- **Dot product:** $\\mathbf{x} \\cdot \\mathbf{y} = \\sum_{i=1}^n x_i y_i$\n- **Norm (length):** $\\|\\mathbf{x}\\| = \\sqrt{\\mathbf{x} \\cdot \\mathbf{x}} = \\sqrt{\\sum_{i=1}^n x_i^2}$\n- **Distance:** $d(\\mathbf{x}, \\mathbf{y}) = \\|\\mathbf{x} - \\mathbf{y}\\|$\n\n### 6.2 Limits and Continuity in $\\mathbb{R}^n$\n\nLet $f: D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}^m$ and let $\\mathbf{a}$ be a limit point of $D$. We say $\\lim_{\\mathbf{x} \\to \\mathbf{a}} f(\\mathbf{x}) = \\mathbf{L}$ if for every $\\varepsilon > 0$, there exists $\\delta > 0$ such that $0< \\|\\mathbf{x} - \\mathbf{a}\\|< \\delta$ implies $\\|f(\\mathbf{x}) - \\mathbf{L}\\|< \\varepsilon$.\n\nThe function $f$ is continuous at $\\mathbf{a}$ if $\\lim_{\\mathbf{x} \\to \\mathbf{a}} f(\\mathbf{x}) = f(\\mathbf{a})$.\n\n## 7. Partial Derivatives\n\nLet $f: D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}$ and let $\\mathbf{a} \\in D$. The partial derivative of $f$ with respect to the $i$-th variable at $\\mathbf{a}$ is:\n$$\\frac{\\partial f}{\\partial x_i}(\\mathbf{a}) = \\lim_{h \\to 0} \\frac{f(a_1, \\dots, a_{i-1}, a_i + h, a_{i+1}, \\dots, a_n) - f(\\mathbf{a})}{h}$$\n\nIf this limit exists for all $i = 1, \\dots, n$, the gradient of $f$ at $\\mathbf{a}$ is:\n$$\\nabla f(\\mathbf{a}) = \\left(\\frac{\\partial f}{\\partial x_1}(\\mathbf{a}), \\frac{\\partial f}{\\partial x_2}(\\mathbf{a}), \\dots, \\frac{\\partial f}{\\partial x_n}(\\mathbf{a})\\right)$$\n\n## 8. Differentiability in $\\mathbb{R}^n$\n\nA function $f: D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}^m$ is differentiable at $\\mathbf{a} \\in D$ if there exists a linear transformation $Df(\\mathbf{a}): \\mathbb{R}^n \\to \\mathbb{R}^m$ such that:\n$$\\lim_{\\mathbf{h} \\to \\mathbf{0}} \\frac{\\|f(\\mathbf{a} + \\mathbf{h}) - f(\\mathbf{a}) - Df(\\mathbf{a})\\mathbf{h}\\|}{\\|\\mathbf{h}\\|} = 0$$\n\nThe linear transformation $Df(\\mathbf{a})$ is called the total derivative of $f$ at $\\mathbf{a}$.\n\nWhen $m = 1$, $Df(\\mathbf{a})$ is represented by the gradient vector $\\nabla f(\\mathbf{a})$, and:\n$$Df(\\mathbf{a})\\mathbf{h} = \\nabla f(\\mathbf{a}) \\cdot \\mathbf{h}$$\n\n**Theorem:** If all partial derivatives of $f$ exist and are continuous in a neighborhood of $\\mathbf{a}$, then $f$ is differentiable at $\\mathbf{a}$.\n\n## 9. The Chain Rule in Multiple Dimensions\n\nLet $f: U \\subseteq \\mathbb{R}^n \\to \\mathbb{R}^m$ be differentiable at $\\mathbf{a} \\in U$, and let $g: V \\subseteq \\mathbb{R}^m \\to \\mathbb{R}^p$ be differentiable at $f(\\mathbf{a}) \\in V$. Then $g \\circ f$ is differentiable at $\\mathbf{a}$, and:\n$$D(g \\circ f)(\\mathbf{a}) = Dg(f(\\mathbf{a})) \\circ Df(\\mathbf{a})$$\n\nIn matrix form, if $f: \\mathbb{R}^n \\to \\mathbb{R}^m$ and $g: \\mathbb{R}^m \\to \\mathbb{R}^p$, then:\n$$J(g \\circ f)(\\mathbf{a}) = Jg(f(\\mathbf{a})) \\cdot Jf(\\mathbf{a})$$\nwhere $Jf(\\mathbf{a})$ is the $m \\times n$ Jacobian matrix of $f$ at $\\mathbf{a}$.\n\n## 10. Directional Derivatives and the Gradient\n\nLet $f: D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}$ and let $\\mathbf{a} \\in D$. For a unit vector $\\mathbf{u} \\in \\mathbb{R}^n$, the directional derivative of $f$ at $\\mathbf{a}$ in the direction $\\mathbf{u}$ is:\n$$D_{\\mathbf{u}} f(\\mathbf{a}) = \\lim_{h \\to 0} \\frac{f(\\mathbf{a} + h\\mathbf{u}) - f(\\mathbf{a})}{h}$$\n\n**Theorem:** If $f$ is differentiable at $\\mathbf{a}$, then for any unit vector $\\mathbf{u}$:\n$$D_{\\mathbf{u}} f(\\mathbf{a}) = \\nabla f(\\mathbf{a}) \\cdot \\mathbf{u}$$\n\nThe gradient points in the direction of maximum increase, and its magnitude is the maximum rate of increase.\n\n## 11. Higher-Order Partial Derivatives and Taylor's Theorem\n\nLet $f: D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}$ be $k$ times differentiable. The $k$-th order partial derivatives are denoted:\n$$\\frac{\\partial^k f}{\\partial x_{i_1} \\partial x_{i_2} \\cdots \\partial x_{i_k}}$$\n\n**Taylor's Theorem (Multivariable):** If $f$ is $(k+1)$ times continuously differentiable on an open convex set $U \\subseteq \\mathbb{R}^n$ containing $\\mathbf{a}$ and $\\mathbf{a} + \\mathbf{h}$, then:\n$$f(\\mathbf{a} + \\mathbf{h}) = \\sum_{|\\alpha| \\leq k} \\frac{D^{\\alpha} f(\\mathbf{a})}{\\alpha!} \\mathbf{h}^{\\alpha} + R_k(\\mathbf{a}, \\mathbf{h})$$\nwhere $\\alpha = (\\alpha_1, \\dots, \\alpha_n)$ is a multi-index with $|\\alpha| = \\alpha_1 + \\dots + \\alpha_n$, $D^{\\alpha} f = \\frac{\\partial^{|\\alpha|} f}{\\partial x_1^{\\alpha_1} \\cdots \\partial x_n^{\\alpha_n}}$, $\\alpha! = \\alpha_1! \\cdots \\alpha_n!$, $\\mathbf{h}^{\\alpha} = h_1^{\\alpha_1} \\cdots h_n^{\\alpha_n}$, and\n$$R_k(\\mathbf{a}, \\mathbf{h}) = \\sum_{|\\alpha| = k+1} \\frac{(k+1)}{\\alpha!} \\mathbf{h}^{\\alpha} \\int_0^1 (1-t)^k D^{\\alpha} f(\\mathbf{a} + t\\mathbf{h})\\,dt$$\n\n## 12. Implicit Function Theorem\n\nLet $F: U \\subseteq \\mathbb{R}^{n+m} \\to \\mathbb{R}^m$ be continuously differentiable on an open set $U$, and write points in $\\mathbb{R}^{n+m}$ as $(\\mathbf{x}, \\mathbf{y})$ where $\\mathbf{x} \\in \\mathbb{R}^n$ and $\\mathbf{y} \\in \\mathbb{R}^m$.\n\nSuppose $(\\mathbf{a}, \\mathbf{b}) \\in U$ satisfies $F(\\mathbf{a}, \\mathbf{b}) = \\mathbf{0}$, and the Jacobian matrix $\\frac{\\partial F}{\\partial \\mathbf{y}}(\\mathbf{a}, \\mathbf{b})$ is invertible.\n\nThen there exist open neighborhoods $A \\subseteq \\mathbb{R}^n$ of $\\mathbf{a}$ and $B \\subseteq \\mathbb{R}^m$ of $\\mathbf{b}$, and a unique continuously differentiable function $g: A \\to B$ such that:\n- $g(\\mathbf{a}) = \\mathbf{b}$\n- $F(\\mathbf{x}, g(\\mathbf{x})) = \\mathbf{0}$ for all $\\mathbf{x} \\in A$\n\nMoreover, the derivative of $g$ is given by:\n$$Dg(\\mathbf{x}) = -\\left(\\frac{\\partial F}{\\partial \\mathbf{y}}(\\mathbf{x}, g(\\mathbf{x}))\\right)^{-1} \\frac{\\partial F}{\\partial \\mathbf{x}}(\\mathbf{x}, g(\\mathbf{x}))$$\n\n## 13. Multiple Integrals\n\n### 13.1 Double Integrals\n\nLet $f: R = [a, b] \\times [c, d] \\to \\mathbb{R}$ be bounded. The double integral of $f$ over $R$ is defined as:\n$$\\iint_R f(x, y)\\,dA = \\lim_{\\|\\Delta\\| \\to 0} \\sum_{i=1}^m \\sum_{j=1}^n f(x_i^*, y_j^*) \\Delta A_{ij}$$\nwhere $\\Delta A_{ij} = \\Delta x_i \\Delta y_j$, and the limit is taken as the mesh of the partition goes to zero.\n\n**Fubini's Theorem:** If $f$ is continuous on $R = [a, b] \\times [c, d]$, then:\n$$\\iint_R f(x, y)\\,dA = \\int_a^b \\int_c^d f(x, y)\\,dy\\,dx = \\int_c^d \\int_a^b f(x, y)\\,dx\\,dy$$\n\n### 13.2 Change of Variables Formula\n\nLet $T: U \\subseteq \\mathbb{R}^n \\to V \\subseteq \\mathbb{R}^n$ be a $C^1$ diffeomorphism (bijective with $C^1$ inverse), and let $f: V \\to \\mathbb{R}$ be continuous. Then:\n$$\\int_V f(\\mathbf{x})\\,d\\mathbf{x} = \\int_U f(T(\\mathbf{u})) |\\det(D"}
{"model": "mlx-community/Qwen3-VL-4B-Instruct-4bit", "status": "OK", "load_time_s": 2, "gen_time_s": 25.5, "prompt_tokens": 27, "completion_tokens": 2180, "total_tokens": 2207, "tokens_per_sec": 85.48, "content_preview": "Certainly. Below is a rigorous, mathematically precise exposition of calculus concepts, progressing from **limits** through **multivariable calculus**, with full formal notation and conceptual foundations.  ---  ## 1. **Limits**  ### Definition (\u03b5-\u03b4 Definition of Limit)  Let $ f: \\mathbb{R} \\to \\mat", "content": "Certainly. Below is a rigorous, mathematically precise exposition of calculus concepts, progressing from **limits** through **multivariable calculus**, with full formal notation and conceptual foundations.\n\n---\n\n## 1. **Limits**\n\n### Definition (\u03b5-\u03b4 Definition of Limit)\n\nLet $ f: \\mathbb{R} \\to \\mathbb{R} $ be a function. We say:\n\n> $$\n\\lim_{x \\to a} f(x) = L\n$$\n\nif for every $ \\varepsilon > 0 $, there exists a $ \\delta > 0 $ such that:\n\n> $$\n\\forall x \\in \\mathbb{R}, \\quad 0< |x - a|< \\delta \\Rightarrow |f(x) - L|< \\varepsilon.\n$$\n\n### Properties of Limits\n\n- **Linearity**:\n  $$\n  \\lim_{x \\to a} [c_1 f(x) + c_2 g(x)] = c_1 \\lim_{x \\to a} f(x) + c_2 \\lim_{x \\to a} g(x),\n  $$\n  provided the limits exist.\n\n- **Product Rule**:\n  $$\n  \\lim_{x \\to a} [f(x) \\cdot g(x)] = \\left( \\lim_{x \\to a} f(x) \\right) \\cdot \\left( \\lim_{x \\to a} g(x) \\right),\n  $$\n  provided the limits exist and are finite.\n\n- **Quotient Rule**:\n  $$\n  \\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\frac{ \\lim_{x \\to a} f(x) }{ \\lim_{x \\to a} g(x) }, \\quad \\text{if } \\lim_{x \\to a} g(x) \\ne 0.\n  $$\n\n- **Squeeze Theorem**:\n  If $ f(x) \\le g(x) \\le h(x) $ for all $ x $ near $ a $, and $ \\lim_{x \\to a} f(x) = \\lim_{x \\to a} h(x) = L $, then $ \\lim_{x \\to a} g(x) = L $.\n\n---\n\n## 2. **Continuity**\n\n> A function $ f: \\mathbb{R} \\to \\mathbb{R} $ is continuous at $ x = a $ if:\n\n$$\n\\lim_{x \\to a} f(x) = f(a).\n$$\n\n### Properties\n\n- **Algebra of Continuous Functions**:\n  If $ f $ and $ g $ are continuous at $ a $, then $ f + g $, $ f \\cdot g $, and $ f/g $ (where $ g(a) \\ne 0 $) are continuous at $ a $.\n\n- **Composites**:\n  If $ f $ is continuous at $ a $, and $ g $ is continuous at $ f(a) $, then $ g \\circ f $ is continuous at $ a $.\n\n---\n\n## 3. **Derivatives**\n\n### Definition (Difference Quotient)\n\nThe derivative of $ f $ at $ a $ is:\n\n> $$\nf'(a) = \\lim_{h \\to 0} \\frac{f(a + h) - f(a)}{h},\n$$\n\nprovided the limit exists.\n\n### General Definition\n\n> $$\nf'(x) = \\lim_{h \\to 0} \\frac{f(x + h) - f(x)}{h}.\n$$\n\n### Rules of Differentiation\n\n- **Power Rule**:\n  $$\n  \\frac{d}{dx} x^n = n x^{n-1}, \\quad n \\in \\mathbb{Z}.\n  $$\n\n- **Product Rule**:\n  $$\n  \\frac{d}{dx} [u(x) v(x)] = u'(x) v(x) + u(x) v'(x).\n  $$\n\n- **Quotient Rule**:\n  $$\n  \\frac{d}{dx} \\left( \\frac{u(x)}{v(x)} \\right) = \\frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}, \\quad v(x) \\ne \ufffd 0.\n  $$\n\n- **Chain Rule**:\n  $$\n  \\frac{d}{dx} f(g(x)) = f'(g(x)) \\cdot g'(x).\n  $$\n\n---\n\n## 4. **Mean Value Theorem (MVT)**\n\nLet $ f: [a, b] \\to \\mathbb{R} $ be continuous on $[a,b]$ and differentiable on $(a,b)$. Then there exists $ c \\in (a,b) $ such that:\n\n> $$\nf'(c) = \\frac{f(b) - f(a)}{b - a}.\n$$\n\n### Generalized Mean Value Theorem\n\nIf $ f $ and $ g $ are continuous on $[a,b]$ and differentiable on $(a,b)$, and $ g'(x) \\ne 0 $ on $ (a,b) $, then there exists $ c \\in (a,b) $ such that:\n\n> $$\n\\frac{f'(c)}{g'(c)} = \\frac{f(b) - f(a)}{g(b) - g(a)}.\n$$\n\n---\n\n## 5. **Fundamental Theorem of Calculus**\n\n### Part I\n\nLet $ f $ be continuous on $[a,b]$, and define $ F(x) = \\int_a^x f(t) \\, dt $.\n\nThen $ F'(x) = f(x) $ for all $ x \\in [a,b] $.\n\n### Part II\n\nLet $ f $ be continuous on $[a,b]$, and $ F $ any antiderivative of $ f $ on $[a,b]$.\n\nThen:\n\n> $$\n\\int_a^b f(x) \\, dx = F(b) - F(a).\n$$\n\n---\n\n## 6. **Multiple Integrals**\n\n### Double Integral\n\nLet $ f: \\mathbb{R}^2 \\to \\mathbb{R} $ be integrable on a region $ D \\subset \\mathbb{R}^2 $. Then:\n\n> $$\n\\iint_D f(x,y) \\, dx \\, dy = \\lim_{\\|\\Delta\\| \\to 0} \\sum_{i=1}^n f(x_i, y_i) \\cdot \\Delta A_i,\n$$\n\nwhere $ \\Delta A_i $ is the area of the $ i $-th subrectangle, and $ (x_i, y_i) $ is a sample point in the subrectangle.\n\n### Fubini\u2019s Theorem\n\nIf $ f $ is continuous on a rectangle $ [a,b] \\times [c,d] $, then:\n\n> $$\n\\iint_{[a,b] \\times [c,d]} f(x,y) \\, dx \\, dy = \\int_a^b \\left( \\int_c^d f(x,y) \\, dy \\right) dx = \\int_c^d \\left( \\int_a^b f(x,y) \\, dx \\right) dy.\n$$\n\n---\n\n## 7. **Vector-Valued Functions and Directional Derivatives**\n\nLet $ \\vec{r}(t) = \\langle x(t), y(t) \\rangle $, $ t \\in \\mathbb{R} $.\n\n- $ \\vec{r}'(t) = \\langle x'(t), y'(t) \\rangle $.\n\n- The **velocity** is $ \\vec{r}'(t) $.\n\n- The **speed** is $ |\\vec{r}'(t)| $.\n\n---\n\n## 8. **Multivariable Differentiation**\n\n### Definition of Partial Derivative\n\nLet $ f: \\mathbb{R}^n \\to \\mathbb{R} $. The partial derivative of $ f $ with respect to $ x_i $ at $ \\vec{x} = (x_1, \\dots, x_n) $ is:\n\n> $$\n\\frac{\\partial f}{\\partial x_i}(\\vec{x}) = \\lim_{h \\to 0} \\frac{f(x_1, \\dots, x_i + h, \\dots, x_n) - f(x_1, \\dots, x_i, \\dots, x_n)}{h}.\n$$\n\n### Gradient\n\n> $$\n\\nabla f(\\vec{x}) = \\left( \\frac{\\partial f}{\\partial x_1}, \\dots, \\frac{\\partial f}{\\partial x_n} \\right).\n$$\n\n### Chain Rule\n\nLet $ \\vec{g}(t) = (g_1(t), \\dots, g_n(t)) $, $ \\vec{f}(\\vec{x}) $, with $ \\vec{x} = (x_1, \\dots, x_n) $.\n\nThen:\n\n> $$\n\\frac{d}{dt} f(\\vec{g}(t)) = \\sum_{j=1}^n \\frac{\\partial f}{\\partial x_j} \\cdot \\frac{d g_j}{dt}.\n$$\n\n---\n\n## 9. **Directional Derivative**\n\nLet $ \\vec{v} \\in \\mathbb{R}^n $ be a unit vector.\n\n> $$\nD_{\\vec{v}} f(\\vec{x}) = \\nabla f(\\vec{x}) \\cdot \\vec{v} = \\sum_{i=1}^n \\frac{\\partial f}{\\partial x_i} v_i.\n$$\n\n---\n\n## 10. **Multivariable Chain Rule (General)**\n\nLet $ \\vec{g}: \\mathbb{R}^m \\to \\mathbb{R}^n $, $ \\vec{f}: \\mathbb{R}^n \\to \\mathbb{R}^p $, and $ \\vec{h} = \\vec{f} \\circ \\vec{g} $.\n\nThen $ \\vec{h} $ is differentiable at $ \\vec{x} $ if $ \\vec{g} $ and $ \\vec{f} $ are differentiable at $ \\vec{x} $ and $ \\vec{g}(x) $.\n\n> $$\nD\\vec{h}(\\vec{x}) = D\\vec{f}(\\vec{g}(\\vec{x})) \\cdot D\\vec{g}(\\vec{x}).\n$$\n\nIn notation for Jacobians:\n\n> $$\n\\frac{\\partial \\vec{h}}{\\partial \\vec{x}} = \\frac{\\partial \\vec{f}}{\\partial \\vec{y}} \\cdot \\frac{\\partial \\vec{y}}{\\partial \\vec{x}}.\n$$\n\n---\n\n## 11. **Implicit Function Theorem**\n\nLet $ F: \\mathbb{R}^{n+m} \\to \\mathbb{R}^m $ be continuously differentiable.\n\nIf $ F(\\vec{a}, \\vec{b}) = 0 $, and $ \\frac{\\partial F}{\\partial \\vec{b}}(\\vec{a}, \\vec{b}) $ is invertible, then there exists a unique $ \\vec{b} = \\vec{g}(\\vec{a}) $, continuously differentiable, in a neighborhood of $ \\vec{a} $.\n\n---\n\n## 12. **Taylor\u2019s Theorem (Multivariable)**\n\nLet $ f $ be $ C^k $ in a neighborhood of $ \\vec{a} \\in \\mathbb{R}^n $. Then:\n\n> $$\nf(\\vec{a} + \\vec{h}) = f(\\vec{a}) + \\nabla f(\\vec{a}) \\cdot \\vec{h} + \\frac{1}{2} \\vec{h}^T H(\\vec{a}) \\vec{h} + \\cdots + \\frac{1}{k!} \\vec{h}^T \\frac{\\partial^k f}{\\partial \\vec{x}^k}(\\vec{a}) \\vec{h} + R_k,\n$$\n\nwhere $ R_k $ is the remainder term.\n\n---\n\n## 13. **Lagrange Multipliers (Optimization)**\n\nLet $ f $ and $ g $ be differentiable. To find local extrema of $ f $ subject to $ g(x) = 0 $, solve:\n\n> $$\n\\nabla f = \\lambda \\nabla g, \\quad g(x) = 0.\n$$\n\n---\n\n## 14. **Vector Calculus (Differential Operators)**\n\n- **Gradient** $ \\nabla f $: directional derivative operator.\n- **Divergence** $ \\nabla \\cdot \\vec{F} $: scalar field measuring \"source\" of a vector field.\n- **Curl** $ \\nabla \\times \\vec{F} $: vector field measuring rotation.\n\n---\n\n## 15. **Stokes\u2019 Theorem**\n\nLet $ \\vec{F} $ be a smooth vector field on a surface $ S $ with boundary $ \\partial S $, oriented by the right-hand rule.\n\n> $$\n\\iint_S (\\nabla \\times \\vec{F}) \\cdot d\\vec{S} = \\oint_{\\partial S} \\vec{F} \\cdot d\\vec{r}.\n$$\n\n---\n\n## 16. **Divergence Theorem (Gauss\u2019s Theorem)**\n\nLet $ \\vec{F} $ be a vector field, $ V $ a solid region with boundary $ \\partial V $, oriented outward.\n\n> $$\n\\iiint_V \\nabla \\cdot \\vec{F} \\, dV = \\iint_{\\partial V} \\vec{F} \\cdot d\\vec{S}.\n$$\n\n---\n\n## 17. **Green\u2019s Theorem (2D Special Case)**\n\nLet $ C $ be a positively oriented, piecewise smooth, closed curve in $ \\mathbb{R}^2 $, $ D $ the region enclosed by $ C $.\n\n> $$\n\\oint_C \\vec{F} \\cdot d\\vec{r} = \\iint_D \\nabla \\cdot \\vec{F} \\, dA.\n$$\n\n---\n\n## Summary\n\n- **Limits** \u2192 **Continuity** \u2192 **Differentiation** \u2192 **Integration** \u2192 **Multivariable** \u2192 **Vector Calculus** \u2192 **Theorems (MVT, FTOC, Stokes, Divergence)**\n\n---\n\n## Key Concepts Recap\n\n| Concept | Notation | Meaning |\n|--------|----------|---------|\n| Limit | $ \\lim_{x \\to a} f(x) = L $ | Behavior of $ f $ near $ a $ |\n| Derivative | $ f'(x) $ | Rate of change of $ f $ |\n| Gradient | $ \\nabla f $ | Direction of steepest ascent |\n| Divergence | $ \\nabla \\cdot \\vec{F} $ | \"Source\" strength |\n| Curl | $ \\nabla \\times \\vec{F} $ | Rotation of vector field |\n| Jacobian | $ D\\vec{f} $ | Derivative of vector-valued function |\n| Chain Rule | $ \\frac{d}{dt} f(g(t)) $ | Composite derivative |\n\n---\n\nThis progression from limits to multivariable calculus provides a rigorous foundation for modern analysis and applied mathematics. The tools developed here form the basis of differential equations, physics, engineering, and numerical analysis.\n\n--- \n\nLet me know if you'd like visualizations, examples, or more advanced topics (e.g., differential forms, manifolds, or PDEs)."}
{"model": "mlx-community/Qwen3-VL-4B-Instruct-8bit", "status": "OK", "load_time_s": 2, "gen_time_s": 34.35, "prompt_tokens": 27, "completion_tokens": 2209, "total_tokens": 2236, "tokens_per_sec": 64.31, "content_preview": "Certainly. Below is a rigorous, step-by-step exposition of calculus concepts, progressing from the foundational notion of **limits** through **single-variable differentiation and integration**, and culminating in **multivariable calculus**, all presented with precise mathematical notation.  ---  ## ", "content": "Certainly. Below is a rigorous, step-by-step exposition of calculus concepts, progressing from the foundational notion of **limits** through **single-variable differentiation and integration**, and culminating in **multivariable calculus**, all presented with precise mathematical notation.\n\n---\n\n## 1. Limits: The Foundation\n\n### Definition (\u03b5-\u03b4):\nLet $ f : \\mathbb{R} \\to \\mathbb{R} $ be a function. We say that\n\n$$\n\\lim_{x \\to a} f(x) = L\n$$\n\nif for every $ \\varepsilon > 0 $, there exists a $ \\delta > 0 $ such that\n\n$$\n\\forall x \\in \\mathbb{R}, \\quad 0< |x - a|< \\delta \\implies |f(x) - L|< \\varepsilon.\n$$\n\nThis is the **\u03b5-\u03b4 definition of a limit**.\n\n### One-Sided Limits:\n- Left-hand limit: $ \\lim_{x \\to a^-} f(x) = L $ if for every $ \\varepsilon > 0 $, there exists $ \\delta > 0 $ such that $ a - \\delta< x< a \\implies |f(x) - L|< \\varepsilon $.\n- Right-hand limit: $ \\lim_{x \\to a^+} f(x) = L $ similarly.\n\n### Limit Laws (for $ \\lim_{x \\to a} f(x) = L $, $ \\lim_{x \\to a} g(x) = M $):\n1. Sum: $ \\lim_{x \\to a} [f(x) + g(x)] = L + M $\n2. Product: $ \\lim_{x \\to a} [f(x)g(x)] = LM $\n3. Quotient: $ \\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\frac{L}{M} $, if $ M \\ne 0 $\n4. Constant Multiple: $ \\lim_{x \\to a} c f(x) = cL $\n5. Power: $ \\lim_{x \\to a} [f(x)]^n = L^n $, for $ n \\in \\mathbb{N} $\n\n---\n\n## 2. Continuity\n\nA function $ f $ is **continuous at $ a $** if:\n\n$$\n\\lim_{x \\to a} f(x) = f(a)\n$$\n\nIf $ f $ is continuous at every point in an interval, it is **continuous on that interval**.\n\n**Intermediate Value Theorem (IVT):** If $ f $ is continuous on $ [a, b] $, and $ y_0 \\in [f(a), f(b)] $, then there exists $ c \\in [a, b] $ such that $ f(c) = y_0 $.\n\n---\n\n## 3. Derivatives: The Rate of Change\n\n### Definition:\nThe **derivative** of $ f $ at $ a $ is:\n\n$$\nf'(a) = \\lim_{h \\to 0} \\frac{f(a + h) - f(a)}{h}\n$$\n\nprovided the limit exists.\n\nIf $ f $ is differentiable at every point in an interval, we write $ f' : I \\to \\mathbb{R} $.\n\n### Derivative Rules:\n\n1. **Power Rule:** If $ f(x) = x^n $, then $ f'(x) = nx^{n-1} $, for $ n \\in \\mathbb{R} $.\n2. **Constant Rule:** $ \\frac{d}{dx}[c] = 0 $\n3. **Sum Rule:** $ \\frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) $\n4. **Product Rule:** $ \\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) $\n5. **Quotient Rule:** $ \\frac{d}{dx}\\left[\\frac{f(x)}{g(x)}\\right] = \\frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} $\n6. **Chain Rule:** If $ y = f(g(x)) $, then $ \\frac{dy}{dx} = f'(g(x)) \\cdot g'(x) $\n\n### Higher-Order Derivatives:\n- $ f''(x) = \\frac{d}{dx}[f'(x)] $, second derivative.\n- In general, $ f^{(n)}(x) $ denotes the $ n $-th derivative.\n\n---\n\n## 4. The Mean Value Theorem (MVT)\n\nLet $ f $ be continuous on $ [a, b] $ and differentiable on $ (a, b) $. Then there exists $ c \\in (a, b) $ such that\n\n$$\nf'(c) = \\frac{f(b) - f(a)}{b - a}\n$$\n\nThis is a key result used in analysis and optimization.\n\n---\n\n## 5. The Fundamental Theorem of Calculus (FTC)\n\n### Part I:\nIf $ f $ is continuous on $ [a, b] $, and $ F(x) = \\int_a^x f(t) \\, dt $, then $ F'(x) = f(x) $.\n\n### Part II:\nIf $ F $ is any antiderivative of $ f $, then\n\n$$\n\\int_a^b f(x) \\, dx = F(b) - F(a)\n$$\n\nThis connects differentiation and integration.\n\n---\n\n## 6. Techniques of Integration\n\n- **Substitution (u-substitution):** If $ u = g(x) $, then $ \\int f(g(x))g'(x) \\, dx = \\int f(u) \\, du $\n- **Integration by Parts:** $ \\int u \\, dv = uv - \\int v \\, du $\n- **Trigonometric Substitution:** For expressions like $ \\sqrt{a^2 - x^2} $, $ \\sqrt{a^2 + x^2} $, or $ \\sqrt{x^2 - a^2} $, use trig identities.\n- **Partial Fractions:** For rational functions, decompose into simpler fractions.\n\n---\n\n## 7. Series and Convergence\n\n### Taylor Series:\nFor $ f $ infinitely differentiable at $ a $, the Taylor series is\n\n$$\nf(x) = \\sum_{n=0}^\\infty \\frac{f^{(n)}(a)}{n!}(x - a)^n\n$$\n\nIf the series converges to $ f(x) $ for all $ x \\in I $, it is called the **Taylor expansion** of $ f $ about $ a $.\n\n### Power Series:\nA series of the form $ \\sum_{n=0}^\\infty a_n (x - a)^n $, converging in an interval $ (a - R, a + R) $, where $ R $ is the radius of convergence.\n\n---\n\n## 8. Multivariable Calculus\n\n### 8.1. Limits and Continuity in $ \\mathbb{R}^n $\n\nLet $ f : \\mathbb{R}^n \\to \\mathbb{R} $, and $ \\mathbf{a} \\in \\mathbb{R}^n $. We say\n\n$$\n\\lim_{\\mathbf{x} \\to \\mathbf{a}} f(\\mathbf{x}) = L\n$$\n\nif for every $ \\varepsilon > 0 $, there exists $ \\delta > 0 $ such that\n\n$$\n\\forall \\mathbf{x} \\in \\mathbb{R}^n, \\quad 0< \\|\\mathbf{x} - \\mathbf{a}\\|< \\delta \\implies |f(\\mathbf{x}) - L|< \\varepsilon\n$$\n\nwhere $ \\|\\cdot\\| $ is the Euclidean norm.\n\n**Continuity in $ \\mathbb{R}^n $** is defined similarly: $ f $ is continuous at $ \\mathbf{a} $ if $ \\lim_{\\mathbf{x} \\to \\mathbf{a}} f(\\mathbf{x}) = f(\\mathbf{a}) $.\n\n---\n\n### 8.2. Partial Derivatives\n\nLet $ f : \\mathbb{R}^n \\to \\mathbb{R} $. The **partial derivative** of $ f $ with respect to $ x_i $ at $ \\mathbf{a} $ is:\n\n$$\n\\frac{\\partial f}{\\partial x_i}(\\mathbf{a}) = \\lim_{h \\to 0} \\frac{f(\\mathbf{a} + h\\mathbf{e}_i) - f(\\mathbf{a})}{h}\n$$\n\nwhere $ \\mathbf{e}_i $ is the unit vector in the $ i $-th coordinate direction.\n\n---\n\n### 8.3. Directional Derivatives\n\nLet $ \\mathbf{v} \\in \\mathbb{R}^n $ be a unit vector. The **directional derivative** of $ f $ at $ \\mathbf{a} $ in direction $ \\mathbf{v} $ is:\n\n$$\nD_{\\mathbf{v}} f(\\mathbf{a}) = \\lim_{h \\to 0} \\frac{f(\\mathbf{a} + h\\mathbf{v}) - f(\\mathbf{a})}{h}\n$$\n\nIt can also be expressed as:\n\n$$\nD_{\\mathbf{v}} f(\\mathbf{a}) = \\nabla f(\\mathbf{a}) \\cdot \\mathbf{v}\n$$\n\nwhere $ \\nabla f(\\mathbf{a}) $ is the gradient vector.\n\n---\n\n### 8.4. Gradient, Divergence, and Curl\n\n- **Gradient** of $ f $: $ \\nabla f(\\mathbf{x}) = \\left( \\frac{\\partial f}{\\partial x_1}, \\dots, \\frac{\\partial f}{\\partial x_n} \\right) $\n- **Divergence** of vector field $ \\mathbf{F} = (F_1, \\dots, F_n) $: $ \\nabla \\cdot \\mathbf{F} = \\sum_{i=1}^n \\frac{\\partial F_i}{\\partial x_i} $\n- **Curl** of $ \\mathbf{F} $ (in $ \\mathbb{R}^3 $): $ \\nabla \\times \\mathbf{F} = \\left( \\frac{\\partial F_3}{\\partial y} - \\frac{\\partial F_2}{\\partial z}, \\frac{\\partial F_1}{\\partial z} - \\frac{\\partial F_3}{\\partial x}, \\frac{\\partial F_2}{\\partial x} - \\frac{\\partial F_1}{\\partial y} \\right) $\n\n---\n\n### 8.5. Chain Rule (Multivariable)\n\nIf $ \\mathbf{y} = f(\\mathbf{x}) $ and $ \\mathbf{x} = g(t) $, then\n\n$$\n\\frac{d}{dt} f(g(t)) = \\nabla f(g(t)) \\cdot \\frac{d\\mathbf{g}}{dt}\n$$\n\n---\n\n### 8.6. Multiple Integrals\n\n#### Double Integral:\n$$\n\\iint_D f(x,y) \\, dA = \\int_a^b \\int_c^d f(x,y) \\, dx \\, dy\n$$\n\nfor a region $ D \\subset \\mathbb{R}^2 $.\n\n#### Triple Integral:\n$$\n\\iiint_E f(x,y,z) \\, dV\n$$\n\nfor a region $ E \\subset \\mathbb{R}^3 $.\n\n#### Change of Variables:\nIf $ \\mathbf{u} = \\mathbf{g}(\\mathbf{x}) $ is a smooth bijection with inverse, then\n\n$$\n\\iint_R f(x,y) \\, dx \\, dy = \\iint_S f(\\mathbf{g}^{-1}(\\mathbf{u})) \\left| \\frac{\\partial(x,y)}{\\partial(u,v)} \\right| \\, du \\, dv\n$$\n\n---\n\n### 8.7. Green\u2019s, Stokes\u2019, and Divergence Theorems\n\n#### Green\u2019s Theorem (2D):\nLet $ C $ be a positively oriented, piecewise-smooth, simple closed curve bounding region $ D $, and $ f, g $ continuously differentiable on $ D $. Then\n\n$$\n\\oint_C (f \\, dx + g \\, dy) = \\iint_D \\left( \\frac{\\partial g}{\\partial x} - \\frac{\\partial f}{\\partial y} \\right) \\, dA\n$$\n\n#### Stokes\u2019 Theorem:\nFor a surface $ S $ with boundary curve $ C $, and vector field $ \\mathbf{F} $, then\n\n$$\n\\oint_C \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_S (\\nabla \\times \\mathbf{F}) \\cdot d\\mathbf{S}\n$$\n\n#### Divergence Theorem:\nFor a region $ V $ with boundary $ S $ oriented outward, and vector field $ \\mathbf{F} $:\n\n$$\n\\iiint_V \\nabla \\cdot \\mathbf{F} \\, dV = \\iint_S \\mathbf{F} \\cdot d\\mathbf{S}\n$$\n\n---\n\n## 9. Advanced Topics (Optional)\n\n### 9.1. Implicit Functions Theorem\n\nIf $ F(x, y) = 0 $, and $ \\frac{\\partial F}{\\partial y} \\ne 0 $ at $ (x_0, y_0) $, then $ y = g(x) $ is a differentiable function near $ x_0 $, and\n\n$$\ng'(x_0) = -\\frac{\\partial F / \\partial x}{\\partial F / \\partial y}\n$$\n\n### 9.2. Lagrange Multipliers\n\nFor constrained optimization: maximize/minimize $ f(x, y) $ subject to $ g(x, y) = c $, then solve\n\n$$\n\\nabla f = \\lambda \\nabla g\n$$\n\n---\n\n## Summary\n\n- **Limits** \u2192 define continuity and convergence.\n- **Derivatives** \u2192 rate of change, local linearity.\n- **Integrals** \u2192 accumulation, area under curves, and inverse of differentiation.\n- **Multivariable calculus** \u2192 extends these ideas to functions of several variables, involving partial derivatives, gradients, and vector calculus.\n\n---\n\nThis framework provides a rigorous, mathematically precise foundation for calculus, suitable for advanced undergraduate or graduate study. The concepts are connected via the **Fundamental Theorem of Calculus**, and their generalizations to higher dimensions are essential in physics, engineering, and applied mathematics.\n\nLet me know if you\u2019d like to dive deeper into any specific topic or proof!"}
{"model": "mlx-community/Qwen3.5-397B-A17B-4bit", "status": "OK", "load_time_s": 67, "gen_time_s": 671.02, "prompt_tokens": 27, "completion_tokens": 3227, "total_tokens": 3254, "tokens_per_sec": 4.81, "content_preview": "The following is a structured exposition of calculus, progressing from the foundational definitions of single-variable analysis to the generalized framework of multivariable and vector calculus.  ---  # 1. Foundations: The Real Number System Calculus is built upon the properties of the real numbers ", "content": "The following is a structured exposition of calculus, progressing from the foundational definitions of single-variable analysis to the generalized framework of multivariable and vector calculus.\n\n---\n\n# 1. Foundations: The Real Number System\nCalculus is built upon the properties of the real numbers $\\mathbb{R}$, specifically the **Completeness Axiom**. This ensures that every non-empty set of real numbers bounded above has a least upper bound (supremum). This property distinguishes $\\mathbb{R}$ from $\\mathbb{Q}$ and allows for the definition of limits.\n\n---\n\n# 2. Limits and Continuity (Single Variable)\n\n### 2.1 The Formal Definition of a Limit\nLet $f: D \\to \\mathbb{R}$ where $D \\subseteq \\mathbb{R}$, and let $c$ be a limit point of $D$. We say that the limit of $f(x)$ as $x$ approaches $c$ is $L$, denoted:\n$$ \\lim_{x \\to c} f(x) = L $$\nif and only if:\n$$ \\forall \\epsilon > 0, \\, \\exists \\delta > 0 \\text{ such that } \\forall x \\in D, \\, 0< |x - c|< \\delta \\implies |f(x) - L|< \\epsilon $$\n\n### 2.2 Continuity\nA function $f: D \\to \\mathbb{R}$ is **continuous** at $c \\in D$ if:\n$$ \\lim_{x \\to c} f(x) = f(c) $$\nFormally:\n$$ \\forall \\epsilon > 0, \\, \\exists \\delta > 0 \\text{ s.t. } |x - c|< \\delta \\implies |f(x) - f(c)|< \\epsilon $$\n\n---\n\n# 3. Differential Calculus (Single Variable)\n\n### 3.1 The Derivative\nLet $f: (a, b) \\to \\mathbb{R}$. The derivative of $f$ at $x \\in (a, b)$ is defined as:\n$$ f'(x) = \\lim_{h \\to 0} \\frac{f(x + h) - f(x)}{h} $$\nprovided the limit exists. If $f$ is differentiable at every point in $(a, b)$, $f$ is said to be differentiable on $(a, b)$.\n\n### 3.2 The Mean Value Theorem\nIf $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then $\\exists \\xi \\in (a, b)$ such that:\n$$ f'(\\xi) = \\frac{f(b) - f(a)}{b - a} $$\n\n---\n\n# 4. Integral Calculus (Single Variable)\n\n### 4.1 The Riemann Integral\nLet $f: [a, b] \\to \\mathbb{R}$ be a bounded function. Let $P = \\{x_0, x_1, \\dots, x_n\\}$ be a partition of $[a, b]$ with $a = x_0< x_1< \\dots< x_n = b$. Let $\\Delta x_i = x_i - x_{i-1}$ and $\\|P\\| = \\max_i \\Delta x_i$.\n\nThe **Riemann Integral** is defined as:\n$$ \\int_a^b f(x) \\, dx = \\lim_{\\|P\\| \\to 0} \\sum_{i=1}^n f(x_i^*) \\Delta x_i $$\nwhere $x_i^* \\in [x_{i-1}, x_i]$.\n\n### 4.2 The Fundamental Theorem of Calculus\n**Part I:** If $f$ is continuous on $[a, b]$ and $F(x) = \\int_a^x f(t) \\, dt$, then:\n$$ F'(x) = f(x) $$\n**Part II:** If $F$ is an antiderivative of $f$ on $[a, b]$, then:\n$$ \\int_a^b f(x) \\, dx = F(b) - F(a) $$\n\n---\n\n# 5. Multivariable Calculus: Foundations\n\nWe extend the domain from $\\mathbb{R}$ to $\\mathbb{R}^n$. A point is denoted $\\mathbf{x} = (x_1, \\dots, x_n)$.\n\n### 5.1 Norms and Metrics\nThe Euclidean norm on $\\mathbb{R}^n$ is defined as:\n$$ \\|\\mathbf{x}\\| = \\sqrt{\\sum_{i=1}^n x_i^2} $$\nThe distance between two points is $\\|\\mathbf{x} - \\mathbf{y}\\|$.\n\n### 5.2 Multivariable Limits\nLet $f: D \\subseteq \\mathbb{R}^n \\to \\mathbb{R}$. We say $\\lim_{\\mathbf{x} \\to \\mathbf{c}} f(\\mathbf{x}) = L$ if:\n$$ \\forall \\epsilon > 0, \\, \\exists \\delta > 0 \\text{ s.t. } 0< \\|\\mathbf{x} - \\mathbf{c}\\|< \\delta \\implies |f(\\mathbf{x}) - L|< \\epsilon $$\n*Note: In $\\mathbb{R}^n$, the limit must exist along all possible paths approaching $\\mathbf{c}$.*\n\n---\n\n# 6. Differentiation in $\\mathbb{R}^n$\n\n### 6.1 Partial Derivatives\nFor a scalar field $f: \\mathbb{R}^n \\to \\mathbb{R}$, the partial derivative with respect to the $i$-th variable is:\n$$ \\frac{\\partial f}{\\partial x_i}(\\mathbf{c}) = \\lim_{h \\to 0} \\frac{f(\\mathbf{c} + h\\mathbf{e}_i) - f(\\mathbf{c})}{h} $$\nwhere $\\mathbf{e}_i$ is the standard basis vector.\n\n### 6.2 The Total Derivative (Fr\u00e9chet Derivative)\nExistence of partial derivatives does not guarantee differentiability in $\\mathbb{R}^n$. A function $f: \\mathbb{R}^n \\to \\mathbb{R}$ is differentiable at $\\mathbf{a}$ if there exists a linear map $A: \\mathbb{R}^n \\to \\mathbb{R}$ such that:\n$$ \\lim_{\\mathbf{h} \\to \\mathbf{0}} \\frac{|f(\\mathbf{a} + \\mathbf{h}) - f(\\mathbf{a}) - A\\mathbf{h}|}{\\|\\mathbf{h}\\|} = 0 $$\nIf such $A$ exists, it is represented by the **Gradient**:\n$$ \\nabla f(\\mathbf{c}) = \\left( \\frac{\\partial f}{\\partial x_1}, \\dots, \\frac{\\partial f}{\\partial x_n} \\right) $$\nsuch that $A\\mathbf{h} = \\nabla f(\\mathbf{c}) \\cdot \\mathbf{h}$.\n\n### 6.3 The Chain Rule\nLet $f: \\mathbb{R}^n \\to \\mathbb{R}$ and $\\mathbf{r}: \\mathbb{R} \\to \\mathbb{R}^n$. The derivative of the composition is:\n$$ \\frac{d}{dt} f(\\mathbf{r}(t)) = \\nabla f(\\mathbf{r}(t)) \\cdot \\mathbf{r}'(t) $$\n\n---\n\n# 7. Multiple Integration\n\n### 7.1 Double and Triple Integrals\nFor a scalar field $f: D \\subseteq \\mathbb{R}^2 \\to \\mathbb{R}$, the double integral over a region $D$ is defined via partitions of area $\\Delta A_i$:\n$$ \\iint_D f(x, y) \\, dA = \\lim_{\\|P\\| \\to 0} \\sum_{i=1}^k f(x_i^*, y_i^*) \\Delta A_i $$\n\n### 7.2 Change of Variables\nTo transform coordinates from $(x, y)$ to $(u, v)$ via a transformation $T(u, v) = (x, y)$, we utilize the Jacobian determinant $J$:\n$$ \\iint_D f(x, y) \\, dx \\, dy = \\iint_S f(x(u, v), y(u, v)) \\left| \\frac{\\partial(x, y)}{\\partial(u, v)} \\right| \\, du \\, dv $$\nwhere the Jacobian is:\n$$ J = \\det \\begin{pmatrix} \\frac{\\partial x}{\\partial u} & \\frac{\\partial x}{\\partial v} \\\\ \\frac{\\partial y}{\\partial u} & \\frac{\\partial y}{\\partial v} \\end{pmatrix} $$\n\n---\n\n# 8. Vector Calculus\n\nWe now consider vector fields $\\mathbf{F}: \\mathbb{R}^n \\to \\mathbb{R}^m$.\n\n### 7.1 Differential Operators\nFor a scalar field $\\phi$ and a vector field $\\mathbf{F} = \\langle P, Q, Z \\rangle$ in $\\mathbb{R}^3$:\n\n1.  **Gradient** (Scalar $\\to$ Vector):\n    $$ \\nabla \\phi = \\left\\langle \\frac{\\partial \\phi}{\\partial x}, \\frac{\\partial \\phi}{\\partial y}, \\frac{\\partial \\phi}{\\partial z} \\right\\rangle $$\n2.  **Divergence** (Vector $\\to$ Scalar):\n    $$ \\nabla \\cdot \\mathbf{F} = \\frac{\\partial P}{\\partial x} + \\frac{\\partial Q}{\\partial y} + \\frac{\\partial Z}{\\partial z} $$\n3.  **Curl** (Vector $\\to$ Vector):\n    $$ \\nabla \\times \\mathbf{F} = \\begin{vmatrix} \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\ \\frac{\\partial}{\\partial x} & \\frac{\\partial}{\\partial y} & \\frac{\\partial}{\\partial z} \\\\ P & Q & Z \\end{vmatrix} $$\n\n### 7.2 The Fundamental Theorems of Vector Calculus\nThese theorems generalize the Fundamental Theorem of Calculus ($\\int_a^b f' = f(b) - f(a)$) to higher dimensions.\n\n1.  **Green's Theorem** (2D):\n    $$ \\oint_C (P \\, dx + Q \\, dy) = \\iint_D \\left( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right) \\, dA $$\n2.  **Stokes' Theorem** (3D Surface):\n    $$ \\oint_{\\partial S} \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_S (\\nabla \\times \\mathbf{F}) \\cdot d\\mathbf{S} $$\n3.  **Divergence Theorem** (3D Volume):\n    $$ \\iiint_V (\\nabla \\cdot \\mathbf{F}) \\, dV = \\oiint_{\\partial V} \\mathbf{F} \\cdot d\\mathbf{S} $$\n\n### 7.3 Generalized Stokes' Theorem\nIn the language of differential forms, all the above theorems are instances of the Generalized Stokes' Theorem:\n$$ \\int_{\\Omega} d\\omega = \\int_{\\partial \\Omega} \\omega $$\nwhere $\\omega$ is a differential form and $d\\omega$ is its exterior derivative."}
{"model": "mlx-community/SmolLM3-3B-4bit", "status": "OK", "load_time_s": 2, "gen_time_s": 13.23, "prompt_tokens": 27, "completion_tokens": 1421, "total_tokens": 1448, "tokens_per_sec": 107.44, "content_preview": "Certainly, let's delve into the concept of limits from single-variable calculus and then transition into multivariable calculus, using rigorous mathematical notation.  ### Single-Variable Calculus: Limits  Given a real-valued function \\( f(x) \\), the limit of \\( f(x) \\) as \\( x \\) approaches a point", "content": "Certainly, let's delve into the concept of limits from single-variable calculus and then transition into multivariable calculus, using rigorous mathematical notation.\n\n### Single-Variable Calculus: Limits\n\nGiven a real-valued function \\( f(x) \\), the limit of \\( f(x) \\) as \\( x \\) approaches a point \\( a \\) is denoted as \\( \\lim_{{x \\to a}} f(x) = L \\) if for every positive real number \\( \\epsilon \\), there exists a positive real number \\( \\delta \\) such that for all \\( x \\), if \\( 0< |x - a|< \\delta \\), then \\( |f(x) - L|< \\epsilon \\).\n\nMathematically:\n\n\\[\n\\lim_{{x \\to a}} f(x) = L \\quad \\text{if} \\quad \\forall \\epsilon > 0, \\exists \\delta > 0, \\quad \\forall x, \\quad |x - a|< \\delta \\implies |f(x) - L|< \\epsilon.\n\\]\n\n**Continuity**:\nA function \\( f(x) \\) is continuous at \\( a \\) if \\( \\lim_{{x \\to a}} f(x) = f(a) \\). This is equivalent to the following three conditions:\n1. \\( f(a) \\) is defined,\n2. \\( \\lim_{{x \\to a}} f(x) \\) exists, and\n3. \\( \\lim_{{x \\to a}} f(x) = f(a) \\).\n\n**Derivatives**:\nThe derivative of \\( f(x) \\) at \\( a \\), denoted \\( f'(a) \\), is defined as:\n\n\\[\nf'(a) = \\lim_{{x \\to a}} \\frac{f(x) - f(a)}{x - a}.\n\\]\n\n**Integrals**:\nThe definite integral of \\( f(x) \\) from \\( a \\) to \\( b \\), denoted \\( \\int_a^b f(x) \\, dx \\), is the number \\( L \\) such that:\n\n\\[\n\\int_a^b f(x) \\, dx = \\lim_{{b \\to c}} \\sum_{k=a}^c f(t_k) \\Delta t, \\quad \\text{where} \\quad t_k = a + k\\Delta t, \\quad \\Delta t = \\frac{c - a}{n}, \\quad n \\in \\mathbb{N} \\text{ is large}.\n\\]\n\nHowever, the most common expression is:\n\n\\[\n\\int_a^b f(x) \\, dx = \\lim_{{b \\downarrow c}} \\sum_{k=a}^{b-\\Delta t} f(t_k) \\Delta t, \\quad t_k = a + k\\Delta t, \\quad \\Delta t = \\frac{b - a}{n}, \\quad n \\in \\mathbb{N} \\text{ is large}.\n\\]\n\n**Fundamental Theorem of Calculus**:\nIf \\( F(x) \\) is defined such that \\( F'(x) = f(x) \\), then:\n\n\\[\n\\int_a^b f(x) \\, dx = F(b) - F(a).\n\\]\n\n### Multivariable Calculus: Limits, Derivatives, and Integrals\n\n#### Partial Derivatives:\n\nGiven a function \\( f(x, y) \\), the partial derivatives with respect to \\( x \\) and \\( y \\) at a point \\( (a, b) \\) are defined as:\n\n\\[\nf_x(a, b) = \\lim_{{x \\to a}} \\frac{f(x, b) - f(a, b)}{x - a},\n\\]\n\n\\[\nf_y(a, b) = \\lim_{{y \\to b}} \\frac{f(a, y) - f(a, b)}{y - b}.\n\\]\n\n**Continuity**:\n\nA function \\( f(x, y) \\) is continuous at a point \\( (a, b) \\) if for every \\( \\epsilon > 0 \\), there exists \\( \\delta > 0 \\) such that if \\( |x - a|< \\delta \\) and \\( |y - b|< \\delta \\), then \\( |f(x, y) - f(a, b)|< \\epsilon \\).\n\n**Gradient**:\n\nThe gradient of \\( f \\) at \\( (a, b) \\), denoted \\( \\nabla f(a, b) \\), is:\n\n\\[\n\\nabla f(a, b) = \\left( f_x(a, b), f_y(a, b) \\right).\n\\]\n\n**Double Integrals**:\n\nGiven a function \\( f(x, y) \\), the double integral over a region \\( R \\) is:\n\n\\[\n\\iint_R f(x, y) \\, dx \\, dy = \\lim_{{\\text{partition} \\ R}} \\sum_{k} f(x_k, y_k) \\Delta x_k \\Delta y_k, \\quad \\text{where} \\quad x_k, y_k \\in R, \\quad \\Delta x_k, \\Delta y_k > 0.\n\\]\n\n**Multiple Integrals**:\n\nThe multiple integral of \\( f(x_1, \\ldots, x_n) \\) over the region \\( R_n \\) is:\n\n\\[\n\\int_{R_n} f(x_1, \\ldots, x_n) \\, dx_1 \\cdots dx_n = \\lim_{{\\text{partition} \\ R_n}} \\sum_{k} f(x_1^k, \\ldots, x_n^k) \\Delta x_1^k \\cdots \\Delta x_n^k, \\quad \\text{where} \\quad x_k^j, y_k^j \\in R_n, \\quad \\Delta x_k^j, \\Delta y_k^j > 0.\n\\]\n\n**Fubini's Theorem**:\n\nIf \\( f(x, y) \\) is integrable over \\( R = [a, b] \\times [c, d] \\), then:\n\n\\[\n\\int_a^b \\int_c^d f(x, y) \\, dy \\, dx = \\int_a^b \\int_c^d f(x, y) \\, dx \\, dy, \\quad \\text{and} \\quad \\int_a^b \\int_c^d f(x, y) \\, dy \\, dx = \\lim_{{\\text{take limits as } b, c \\to \\infty \\text{ or } -\\infty}} \\int_a^b \\int_c^d f(x, y) \\, dy \\, dx.\n\\]\n\n**Triple Integrals** and **Higher-Order Integrals**:\n\nThe triple integral of \\( f(x, y, z) \\) over the region \\( R_3 = [a, b] \\times [c, d] \\times [e, f] \\) is:\n\n\\[\n\\iint_{R_3} f(x, y, z) \\, dx \\, dy \\, dz = \\lim_{{\\text{partition} \\ R_3}} \\sum_{k_1, k_2, k_3} f(x_{k_1}, y_{k_2}, z_{k_3}) \\Delta x_{k_1} \\Delta y_{k_2} \\Delta z_{k_3}, \\quad \\text{where} \\quad x_{k_i}, y_{k_i}, z_{k_i} \\in R_3, \\quad \\Delta x_{k_i}, \\Delta y_{k_i}, \\Delta z_{k_i} > 0.\n\\]\n\n**Higher-Order Integrals**:\n\nThe \\( n \\)-fold integral of \\( f(x_1, \\ldots, x_n) \\) over the region \\( R_n = [a, b] \\times [c, d] \\times \\cdots, [q_1, r_1] \\times [q_2, r_2] \\times \\cdots \\) is:\n\n\\[\n\\int_{R_n} f(x_1, \\ldots, x_n) \\, dx_1 \\cdots dx_n = \\lim_{{\\text{partition} \\ R_n}} \\sum_{k_1, k_2, \\ldots, k_n} f(x_{k_1}, \\ldots, x_{k_n}) \\Delta x_{k_1} \\cdots \\Delta x_{k_n}, \\quad \\text{where} \\quad x_{k_i}, y_{k_i}, z_{k_i}, \\ldots, z_{k_n} \\in R_n, \\quad \\Delta x_{k_i}, \\Delta y_{k_i}, \\Delta z_{k_i}, \\ldots, \\Delta z_{k_n} > 0.\n\\]\n\n**Leibniz Rule**:\n\nIf \\( f(x, y) \\) is integrable over \\( [a, b] \\times [c, d] \\), then:\n\n\\[\n\\int_a^b \\int_c^d f(x, y) \\, dy \\, dx = \\int_a^b \\int_c^d \\left( \\int_c^d \\frac{\\partial}{\\partial y} f(x, y) \\, dy + \\frac{\\partial}{\\partial x} f(x, y) \\right) \\, dy \\, dx,\n\\]\n\n**Higher Dimensions**:\n\nThe multivariable chain rule:\n\n\\[\n\\frac{\\partial}{\\partial x} f(x_1, \\ldots, x_n) = \\sum_{i=1}^n \\frac{\\partial f}{\\partial x_i} \\cdot \\frac{\\partial x_i}{\\partial x},\n\\]\n\n**Multiple Integration**:\n\nThe multiple integral of \\( f(x_1, \\ldots, x_n) \\) over the region \\( R_n \\) is:\n\n\\[\n\\int_{R_n} f(x_1, \\ldots, x_n) \\, dx_1 \\cdots dx_n = \\lim_{{\\text{partition} \\ R_n}} \\sum_{k_1, k_2, \\ldots, k_n} f(x_{k_1}, \\ldots, x_{k_n}) \\Delta x_{k_1} \\cdots \\Delta x_{k_n}, \\quad \\text{where} \\quad x_{k_i}, y_{k_i}, z_{k_i}, \\ldots, z_{k_n} \\in R_n, \\quad \\Delta x_{k_i}, \\Delta y_{k_i}, \\Delta z_{k_i}, \\ldots, \\Delta z_{k_n} > 0.\n\\]\n\nThis is a rigorous mathematical framework for understanding limits, derivatives, and integrals in multivariable calculus."}