Updated several sections.

docs/en/mathematics/multivariable-calculus/extrema.md

@@ -77,7 +77,7 @@ will be added later.
 
 ### The general case
 
-*Theorem*: Let $f: S \to \mathbb{R}$ and $\mathbf{g}: \mathbb{R}^m \to \mathbb{R}^n$ with $m$ restrictions given by
+*Theorem*: Let $f: S \to \mathbb{R}$ and $\mathbf{g}: \mathbb{R}^m \to \mathbb{R}^n$ with $m \leq n -1$ restrictions given by
 
 $$
     S := \big\{\mathbf{x} \in \mathbb{R}^n \;\big|\; \mathbf{g}(\mathbf{x}) = 0  \big\} \subseteq D,

docs/en/mathematics/multivariable-calculus/integration.md

@@ -15,12 +15,12 @@ will be added later.
 </details>
 <br>
 
-## Iteration of integralss
+## Iteration of integrals
 
 *Theorem*: for $D \subseteq \mathbb{R}^n$ ($n=2$ for simplicity) bounded and piecewise smooth boundary, let $f: D \to \mathbb{R}$ be bounded and continuous. Let $R$ be a rectangle with $D \subseteq R$ then
 
 $$
-    \iint_D f dA = \iint_R F dA, \qquad \text{where } F(\mathbf{x}) = \begin{cases} F(\mathbf{x}) \quad \mathbf{x} \in D, \\ 0 \quad \mathbf{x} \notin D. \end{cases}
+    \iint_D f dA = \iint_R F dA, \qquad \text{where } F(\mathbf{x}) = \begin{cases} F(\mathbf{x}) \quad &\mathbf{x} \in D, \\ 0 \quad &\mathbf{x} \notin D. \end{cases}
 $$
 
 <details>

docs/en/mathematics/ordinary-differential-equations/second-order-ode.md

@@ -118,7 +118,7 @@ named the Wronskian and we can solve for $c_1(t)$ and $c_2(t)$ by integration.
 
 #### Ansatz method
 
-Let $f(t) = p(t)e^{\lambda t}$, rule of thumb: $y_p$ is of related type to inhomogeneity $f$. Then for $A_n, B_n and P_n$ polynomials of degree $\leq n$ and $\alpha \in \mathbb{R}$
+Let $f(t) = p(t)e^{\lambda t}$, rule of thumb: $y_p$ is of related type to inhomogeneity $f$. Then for $A_n, B_n$ and $P_n$ polynomials of degree $\leq n$ and $\alpha \in \mathbb{R}$
 
 | Inhomogeneity | Particular solution |
 | ------ | --------------- |

docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md

@@ -72,4 +72,4 @@ $$
 \end{align*}
 $$
 
-Demanding that: $Y(t) \mathbf{\dot c}(t) = \mathbf{f}(t)$. Then $\mathbf{\dot c}(t) = Y^{-1}(t) \mathbf{f}(t) \iff Y(t)$ is nonsingular. Then solve for $\mathbf{c}(t)$. 
+Demanding that: $Y(t) \mathbf{\dot c}(t) = \mathbf{f}(t)$ is the Wronskian. Then $\mathbf{\dot c}(t) = Y^{-1}(t) \mathbf{f}(t) \iff Y(t)$ is nonsingular. Then solve for $\mathbf{c}(t)$.