Formatting Fix

docs/math/abstract-algebra/16-rings.md

@@ -1,4 +1,5 @@
 # Chapter 16 - Rings
+
 ## Section 16.1 - Rings
 
 **Definition**. A nonempty set $S$ is a *ring* if, with two binary operations called addition and multipllication, the following are satisfied:
@@ -139,4 +140,4 @@ $$
 
 Thus, $ab \in P$. By symnetry, assume $a \notin P$. Thus, $b \in P$ by the  devinition of a prime ideal, so $b + P = 0 + P$, meaning $R/P$ is an integral domain.
 
-**Theorem**. 16.40: In a commutative ring with identity, every maximal ideal is also a prime ideal.
+**Theorem**. 16.40: In a commutative ring with identity, every maximal ideal is also a prime ideal.

docs/math/abstract-algebra/17-polynomial-rings.md

@@ -1,4 +1,5 @@
 # Chapter 17 - Polynomial Rings
+
 ## Section 17.1 - Polynomial Rings
 
 Throughout this chapter, we will assume that $R$ is a commutative ring with identity.
@@ -63,4 +64,4 @@ Then, if $p | a_i$ for $0 \leq i < n$, but $p \nmid a_n$ and $p^2 \nmid a_0$, th
 
 **Theorem**. If $F$ is a field, then every ideal in $F[x]$ is a principal ideal.
 
-**Theorem**. Let $F$ be a field, and suppose $p(x) \in F[x]$. Then, the ideal $<p(x)>$ is maximal if and only if $p(x)$ is irreducible.
+**Theorem**. Let $F$ be a field, and suppose $p(x) \in F[x]$. Then, the ideal $<p(x)>$ is maximal if and only if $p(x)$ is irreducible.

docs/math/abstract-algebra/18-integral-domains.md

@@ -98,4 +98,4 @@ As a direct consequence, we see the following.
 
 1. Given a field $F$, since $F$ is a PID, it is also a UFD. Thus, $F[x]$ is a UFD.
 2. The ring of polynomials over integers, $\mathbb{Z}[x]$ is a UFD.
-3. Given $D$ is a UFD, $D[x]$ is a UFD. Thus, $D[x_1, x_2]$ is a UFD, and by induction, $D[x_1, \ldots, x_n]$ is a UFD.
+3. Given $D$ is a UFD, $D[x]$ is a UFD. Thus, $D[x_1, x_2]$ is a UFD, and by induction, $D[x_1, \ldots, x_n]$ is a UFD.

docs/math/diffeq/1-intro.md

@@ -36,4 +36,4 @@ Note that $a_n(t)$ does not depeond on any derivative of $y$, so the presence of
 
 This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/DirectionFields.aspx).
 
-**Definition**. A directional field is the graph of a $t$ vs. $y(t)$, with vectors drawn at each point with a slope corresponding to $y'(t)$. Notably, each arrow will be pointed right (towards increasing $t$).
+**Definition**. A directional field is the graph of a $t$ vs. $y(t)$, with vectors drawn at each point with a slope corresponding to $y'(t)$. Notably, each arrow will be pointed right (towards increasing $t$).

docs/math/diffeq/2-1st-order.md

@@ -144,4 +144,4 @@ $$
 \frac{1}{1-n}v' + p(x)v = q(x)
 $$
 
-After solving, be sure to rewrite in terms of $y$.
+After solving, be sure to rewrite in terms of $y$.

docs/math/diffeq/3-2nd-order.md

@@ -32,7 +32,7 @@ Thus, we allow the *charactaristic equation* of the differential equation to be
 
 $$ ar^2 + br + c = 0 $$
 
-# Section 3.2 - Real & Distinct Roots
+## Section 3.2 - Real & Distinct Roots
 
 This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx).
 
@@ -42,11 +42,11 @@ $$ y_1(t) = e^{r_1 t} $$
 
 $$ y_2(t) = e^{r_2 t} $$
 
-Thus, 
+Thus,
 
 $$ y(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t} $$
 
-# Section 3.3 - Complex Roots
+## Section 3.3 - Complex Roots
 
 This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx).
 
@@ -120,7 +120,7 @@ Assume we have the differential equation as follows:
 
 $$ y'' + p(t) y' + q(t) y = g(t) $$
 
-The equivilent homogenous differential equation is 
+The equivilent homogenous differential equation is
 
 $$ y'' + p(t) y' + q(t) y = 0 $$
 
@@ -156,7 +156,7 @@ Assume we have the differential equation as follows:
 
 $$ y'' + p(t) y' + q(t) y = g(t) $$
 
-The equivilent homogenous differential equation is 
+The equivilent homogenous differential equation is
 
 $$ y'' + p(t) y' + q(t) y = 0 $$
 
@@ -164,4 +164,4 @@ For this method, we must have $y_1(t)$ and $y_2(t)$ known. Through a lot of math
 
 $$
 y_p = -y_1 \int \frac{y_2(t)g(t)}{W(y_1, y_2)} dt + y_2 \int \frac{y_1(t)g(t)}{W(y_1, y_2)} dt
-$$
+$$

docs/math/diffeq/4-laplace.md

@@ -1,4 +1,5 @@
 # Section 4 - Laplace Transformations
+
 ## Section 4.1 - Definition
 
 This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx).
@@ -78,4 +79,4 @@ We can take the Laplace transformation of an IVP, solve for $Y(s)$, then take th
 
 ## Section 4.6 - Nonconstant Coefficient IVPs
 
-This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/IVPWithNonConstantCoefficient.aspx).
+This section is from [Paul's Online Math Notes](https://tutorial.math.lamar.edu/Classes/DE/IVPWithNonConstantCoefficient.aspx).