Fix hboxes in vol4

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

ib/ca/03_more_integration.tex

@@ -9,8 +9,10 @@ \subsection{Winding numbers}
 	\]
 \end{definition}
 If \( \gamma \) is a closed curve, \( I(\gamma;w) \) is an integer.
-This is because \( \gamma(a) = \gamma(b) \) implies \( \exp(i\theta(b) - i\theta(a)) = 1 \).
-If \( \theta_1 \colon [a,b] \to \mathbb C \) is also continuous such that \( \gamma(t) = w + re^{i\theta_1(t)} \), then \( \exp(i\theta(t) - i\theta_1(t)) = 1 \), so \( \frac{\theta_1(t) - \theta(t)}{2\pi} \in \mathbb Z \).
+This is because
+\[ \gamma(a) = \gamma(b) \implies \exp(i\theta(b) - i\theta(a)) = 1 \]
+If \( \theta_1 \colon [a,b] \to \mathbb C \) is also continuous such that \( \gamma(t) = w + re^{i\theta_1(t)} \), then \( \exp(i\theta(t) - i\theta_1(t)) = 1 \), so
+\[ \frac{\theta_1(t) - \theta(t)}{2\pi} \in \mathbb Z \]
 Since \( \theta_1 - \theta \) is continuous, this quotient must be a constant.
 Hence, \( I(\gamma;w) \) is well-defined and independent of the (continuous) choice of \( \theta \).
 \begin{lemma}

ib/ca/04_singularities.tex

@@ -87,7 +87,7 @@ \subsection{Poles}
 	If \( a \in U \) is an isolated singularity of \( f \) that is not removable or a pole, it is an \textit{essential singularity}.
 \end{definition}
 \begin{remark}
-	An equivalent characterisation for \( a \) to be an essential singularity is that \( \lim_{z \to a} \abs{f(z)} \) does not exist.
+	An equivalent characterisation for \( a \) to be an essential singularity is that the limit \( \lim_{z \to a} \abs{f(z)} \) does not exist.
 	This follows from the previous proposition and the definition of a pole.
 \end{remark}
 \begin{example}
@@ -172,7 +172,9 @@ \subsection{Laurent series}
 When \( a \) is an essential singularity, we can still obtain an analogous series expansion with infinitely many terms with negative powers.
 More generally, we have the following.
 \begin{theorem}[Laurent expansion]
-	Let \( f \) be holomorphic on an annulus \( A = \qty{z \in \mathbb C \colon r < \abs{z-a} < R} \) for \( 0 \leq r < R \leq \infty \).
+	Let \( f \) be holomorphic on an annulus
+	\[ A = \qty{z \in \mathbb C \colon r < \abs{z-a} < R} \]
+	for \( 0 \leq r < R \leq \infty \).
 	Then:
 	\begin{enumerate}
 		\item \( f \) has a unique convergent series expansion

ib/geom/04_geometry_of_surfaces.tex

@@ -294,9 +294,12 @@ \subsection{Area}
 \subsection{Second fundamental form}
 Let \( \sigma \colon V \to U \subseteq \Sigma \) be allowable.
 By using Taylor's theorem, we can write
-\[
-	\sigma(u+h,v+\ell) = \sigma(u,v) + h \sigma_u(u,v) + \ell \sigma_v(u,v) + \frac{1}{2} \qty(h^2 \sigma_{uu}(u,v) + 2h\ell \sigma_{uv}(u,v) + \ell^2 \sigma_{vv}(u,v)) + O(h^3,\ell^3)
-\]
+\begin{align*}
+	\sigma(u+h,v+\ell) &= \sigma(u,v) \\
+	&+ h \sigma_u(u,v) + \ell \sigma_v(u,v) \\
+	&+ \frac{1}{2} \qty(h^2 \sigma_{uu}(u,v) + 2h\ell \sigma_{uv}(u,v) + \ell^2 \sigma_{vv}(u,v)) \\
+	&+ O(h^3,\ell^3)
+\end{align*}
 where \( h,\ell \) are small, and \( (u+h,v+\ell) \in V \).
 Recall that if \( p = \sigma(u,v) \), we have \( T_p \Sigma = \genset{\qty{\sigma_u,\sigma_v}} \).
 Hence, the orthogonal distance from \( \sigma(u+h,v+\ell) \) to the affine tangent plane \( T_p \Sigma + p \) is given by projection to the normal direction.
@@ -342,9 +345,11 @@ \subsection{Second fundamental form}
 		0 = \inner{n_u, \sigma_u} + \inner{n, \sigma_{uu}} = \inner{n_v, \sigma_v} + \inner{n, \sigma_{vv}} = \inner{n_v, \sigma_u} + \inner{n,\sigma_{uv}}
 	\]
 	Some of these terms appear in the definition of the second fundamental form:
-	\[
-		L = \inner{n,\sigma_{uu}} = -\inner{n_u, \sigma_u};\quad M = \inner{n,\sigma_{uv}} = -\inner{n_v, \sigma_u} = -\inner{n_u, \sigma_v};\quad N = \inner{n,\sigma_{vv}} = -\inner{n_v, \sigma_v}
-	\]
+	\begin{align*}
+		L &= \inner{n,\sigma_{uu}} = -\inner{n_u, \sigma_u} \\
+		M &= \inner{n,\sigma_{uv}} = -\inner{n_v, \sigma_u} = -\inner{n_u, \sigma_v} \\
+		N &= \inner{n,\sigma_{vv}} = -\inner{n_v, \sigma_v}
+	\end{align*}
 	If the second fundamental form vanishes, then \( n_u \) is orthogonal to \( \sigma_u \), \( \sigma_v \), and \( n \) itself.
 	Since \( \sigma_u, \sigma_v, n \) form a basis for \( \mathbb R^3 \), we have \( n_u = 0 \).
 	Similarly, \( n_v = 0 \), hence \( n \) is constant by the mean value theorem.
@@ -465,7 +470,7 @@ \subsection{Gauss maps}
 \end{remark}
 \begin{lemma}
 	The derivative of the Gauss map is self-adjoint.
-	More precisely, viewing \( D \eval{n}_p \colon T_p \Sigma \to T_p \Sigma \) as an endomorphism over the inner product space with the first fundamental form, this linear map satisfies
+	More precisely, viewing the map \( D \eval{n}_p \colon T_p \Sigma \to T_p \Sigma \) as an endomorphism over the inner product space with the first fundamental form, this linear map satisfies
 	\[
 		\Iff_p\qty(D \eval{n}_p(v), w) = \Iff_p\qty(v, D \eval{n}_p(w))
 	\]
@@ -665,9 +670,10 @@ \subsection{Elliptic, hyperbolic, and parabolic points}
 		\abs{\kappa(u,v)} \in \qty(\abs{\kappa(p)}-\varepsilon, \abs{\kappa(p)}+\varepsilon)
 	\]
 	Hence,
-	\[
-		\qty(\abs{\kappa(p)}-\varepsilon) \int_{V_i} \norm{\sigma_u \times \sigma_v} \dd{u}\dd{v} \leq \int_{V_i} \abs{\kappa(u,v)} \cdot \norm{\sigma_u \times \sigma_v} \dd{u}\dd{v} \leq \qty(\abs{\kappa(p)}+\varepsilon) \int_{V_i} \norm{\sigma_u \times \sigma_v} \dd{u}\dd{v}
-	\]
+	\begin{align*}
+		\qty(\abs{\kappa(p)}-\varepsilon) \int_{V_i} \norm{\sigma_u \times \sigma_v} \dd{u}\dd{v} &\leq \int_{V_i} \abs{\kappa(u,v)} \cdot \norm{\sigma_u \times \sigma_v} \dd{u}\dd{v} \\
+		&\leq \qty(\abs{\kappa(p)}+\varepsilon) \int_{V_i} \norm{\sigma_u \times \sigma_v} \dd{u}\dd{v}
+	\end{align*}
 	In other words,
 	\[
 		\abs{\kappa(p)}-\varepsilon \leq \frac{\mathrm{area}_{S^2} (n(A_i))}{\mathrm{area}_{\Sigma} (A_i)} \leq \abs{\kappa(p)} + \varepsilon

ib/geom/06_riemannian_metrics.tex

@@ -37,7 +37,7 @@ \subsection{Definitions}
 			F & G
 		\end{pmatrix}
 	\]
-	So \( Df \) defines an isometry from an open set in the chart \( (U, \varphi(U) = V) \) to one in \( \qty(\widetilde U, \widetilde \varphi\qty(\widetilde U) = \widetilde V) \).
+	So \( Df \) defines an isometry from an open set in the chart \( (U, \varphi(U) = V) \) to one in the chart \( \qty(\widetilde U, \widetilde \varphi\qty(\widetilde U) = \widetilde V) \).
 \end{definition}
 This compatibility condition is the transition law for first fundamental forms for smooth surfaces in \( \mathbb R^3 \).
 \begin{example}

ib/grm/10_algebraic_integers.tex

@@ -128,7 +128,7 @@ \subsection{Algebraic integers}
 \end{definition}
 \begin{corollary}
 	All minimal polynomials are irreducible.
-	By the isomorphism theorem, \( \faktor{\mathbb Q[X]}{(f)} \cong \mathbb Q[\alpha] \leq \mathbb C \).
+	By the first isomorphism theorem, \( \faktor{\mathbb Q[X]}{(f)} \cong \mathbb Q[\alpha] \leq \mathbb C \).
 	Any subring of a field is an integral domain.
 	Hence \( (f) \) is a prime ideal in \( \mathbb Q[X] \), and hence \( f \) is irreducible.
 	In particular, this implies that \( \mathbb Q[\alpha] \) is a field.

ib/grm/12_modules.tex

@@ -461,7 +461,7 @@ \subsection{The structure theorem}
 	The \( d_i \) are called invariant factors.
 \end{theorem}
 \begin{proof}
-	Since \( n \) is finitely generated, there exists a surjective \( R \)-module homomorphism \( \varphi \colon R^m \to M \) for some \( m \).
+	Since \( M \) is a finitely generated module, there exists a surjective \( R \)-module homomorphism \( \varphi \colon R^m \to M \) for some \( m \).
 	By the first isomorphism theorem, \( M \cong \faktor{R^m}{\ker \varphi} \).
 	By the previous theorem, there exists a free basis \( x_1, \dots, x_m \) for \( R^m \) such that \( \ker \varphi \leq R^m \) is generated by \( d_1 x_1, \dots, d_t x_t \) and where \( d_1 \mid \dots \mid d_t \).
 	Then,

todo.txt

@@ -27,6 +27,5 @@ Create a version for print.
 - Remove colour. Search for words referencing colour in diagrams (e.g. "red") and replace text with suitable alternatives.
 - Spellcheck (unless it gets too cumbersome to program one to allow mathematical jargon).
 - Edit the introduction text.
-- Add course introductions.
 - Based on font choices, edit spacing inside \faktor.
 - Fix appearance of \not, such as \not\to (this specific instance has been corrected).