Nabla Operand • Gradient • Divergence • Curl • Laplacian.md

The nabla symbol, denoted as $\nabla$, represents a vector differential operator commonly used in vector calculus. It is used to describe various operations involving vectors, such as gradients, divergences, and curls.

Here are some key operations associated with the nabla operator:

1. Gradient ($\nabla f$):

   • The gradient of a scalar function $f$, results in a vector:$$\nabla f = \frac{\partial f}{\partial x}\hat{i} + \frac{\partial f}{\partial y}\hat{j} +\frac{\partial f}{\partial z}\hat{k}$$~Example: If $f(x, y, z) = x^2 + y^2 + z^2$, then $\nabla f = , <2x ,\ 2y\mathbf ,\ 2z>$.

2. Divergence ($\nabla \cdot \vec F$):

   • The divergence of a vector field $\vec F = (F_x, F_y, F_z)$ is given by $$\nabla \cdot \vec F = {\partial F_x \over \partial x} + {\partial F_y \over \partial y} + {\partial F_z \over \partial z}$$~Example: If $F = (2x, y^2, z)$, then $\nabla \cdot \vec F = 2 + 2y + 1 = 2y + 3$.

3. Curl ($\nabla \times \vec F$):

   • The curl of a vector field $\vec F = (F_x, F_y, F_z)$ is given by:$$\nabla \times \vec F = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right)\hat{i} + \left( \frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x} \right)\hat{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right)\hat{k}$$*~Example: If $F = (2, , y, , 3z + 2x)$, then $\nabla \times F = (0-0)\hat{i} + (0-1)\hat{j} + (0-0)\hat{k} = -2\hat{i}$.

4. Laplacian ($\nabla^2 f$):

   • The Laplacian of a scalar function $f$ is given by:$$ \nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$$~Example: