diff --git a/src/content/2.1/declarative-programming.tex b/src/content/2.1/declarative-programming.tex
index d20ad02d7..a52cd2879 100644
--- a/src/content/2.1/declarative-programming.tex
+++ b/src/content/2.1/declarative-programming.tex
@@ -66,8 +66,8 @@
 These are the direct encodings of the differential equations
 corresponding to Newton's laws of motion:
 \begin{align*}
-  F & = m \frac{dv}{dt} \\
-  v & = \frac{dx}{dt}
+  F & = m \frac{\mathrm{d}v}{\mathrm{d}t} \\
+  v & = \frac{\mathrm{d}x}{\mathrm{d}t}
 \end{align*}
 Similar methods may be applied to more complex problems, like the
 propagation of electromagnetic fields using Maxwell's equations, or even
@@ -93,7 +93,7 @@
 ray refract at the boundary of air and water, resulting in Snell's law
 of refraction:
 \begin{equation*}
-  \frac{sin(\theta_1)}{sin(\theta_2)} = \frac{v_1}{v_2}
+  \frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2}
 \end{equation*}
 where $v_1$ is the speed of light in the air and $v_2$ is
 the speed of light in the water.
@@ -111,7 +111,7 @@
 the total energy). When you fire a mortar to hit a given target, the
 projectile will first go up, where the potential energy due to gravity
 is higher, and spend some time there racking up negative contribution to
-the action. It will also slow down at the top of the parabola, to
+the action. It will also slow down toward the top of the parabola, to
 minimize kinetic energy. Then it will speed up to go quickly through the
 area of low potential energy.
 
diff --git a/src/content/2.2/limits-and-colimits.tex b/src/content/2.2/limits-and-colimits.tex
index 997d1e932..3cbdc68a7 100644
--- a/src/content/2.2/limits-and-colimits.tex
+++ b/src/content/2.2/limits-and-colimits.tex
@@ -461,7 +461,7 @@ \section{Examples of Limits}
 $a$ could, for instance, be the set of three-dimensional vectors,
 and $f$ the vector length. Then the pullback is the set of pairs
 $(v, ())$, where $v$ is a vector of length 1.23 (a
-solution to the equation $\sqrt{(x^{2}+y^{2}+z^{2})} = 1.23$), and
+solution to the equation $\sqrt{x^{2}+y^{2}+z^{2}} = 1.23$), and
 $()$ is the dummy element of the singleton set.
 
 But pullbacks have more general applications, also in programming. For