Fix hboxes in vol2

Signed-off-by: zeramorphic <[email protected]>

ia/analysis/10_integration.tex

@@ -151,7 +151,10 @@ \subsection{Monotonic and continuous functions}
 \end{theorem}
 \begin{proof}
 	Suppose \(f\) is increasing.
-	Then \(\sup_{x \in [x_{j-1} - x_j]} f(x) = f(x_j)\), and similarly, \(\inf_{x \in [x_{j-1} - x_j]} f(x) = f(x_{j-1})\).
+	Then
+	\[\sup_{x \in [x_{j-1} - x_j]} f(x) = f(x_j)\]
+	and similarly
+	\[\inf_{x \in [x_{j-1} - x_j]} f(x) = f(x_{j-1})\]
 	Thus,
 	\[
 		S(f, \mathcal D) - s(f, \mathcal D) = \sum_{j=1}^n (x_j - x_{j-1}) \left[ f(x_j) - f(x_{j-1}) \right]
@@ -202,7 +205,7 @@ \subsection{Monotonic and continuous functions}
 		\mathcal D = \qty{ a, a + \frac{b-a}{n}, a + 2\frac{b-a}{n} + \dots + b }
 	\]
 	where \(n\) is chosen large enough such that \(\frac{b-a}{n} < \delta\).
-	Then, for any \(x, y \in [x_{j-1}, x_j]\), \(\abs{f(x) - f(y)} < \varepsilon\).
+	Then, for any \(x, y \in [x_{j-1}, x_j]\), we have that \(\abs{f(x) - f(y)} < \varepsilon\).
 	We can now write
 	\[
 		\max_{x \in [x_{j-1}, x_j]} f(x) - \min_{x \in [x_{j-1}, x_j]} f(x) = f(p) - f(q) < \varepsilon

ib/opt/04_linear_programming.tex

@@ -146,7 +146,7 @@ \subsection{Basic solutions and basic feasible solutions}
 \begin{enumerate}[A:]
 	\setcounter{enumi}{2}
 	\item Every basic solution has \textit{exactly} \( m \) nonzero entries.
-	      This is known as the non-degeneracy assumption.
+	      This assumption is known as the non-degeneracy assumption.
 	      This assumption cannot be created without loss of generality, but it is far simpler to discuss problems with this assumption met.
 	      Throughout this course, we will keep this assumption to be true.
 \end{enumerate}

ib/vp/04_extensions_to_the_euler_lagrange_equation.tex

@@ -122,7 +122,7 @@ \subsection{Geodesics on surfaces}
 	\Sigma = \qty{ \vb x \colon g(\vb x) = 0 }
 \]
 Consider two points \( A, B \) on \( \Sigma \).
-What are the geodesics (shortest paths on the surface) between the two points, if one exists at all?
+What are the geodesics (the shortest paths on the surface) between the two points, if one exists at all?
 Consider a parametrisation of such a path given by \( t \in [0, 1] \) where \( A = \vb x(0), B = \vb x(1) \).
 We wish to extremise
 \[