Volume 1 fixes

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

ia/analysis/01_limits_and_convergence.tex

@@ -86,7 +86,7 @@ \subsection{Limits in the complex plane}
 
 However, property (vii) makes no sense in the world of the complex numbers since they do not have an ordering.
 
-\subsection{The Bolzano-Weierstrass theorem}
+\subsection{The Bolzano--Weierstrass theorem}
 \begin{theorem}
 	If \(x_n\) is a sequence of real numbers, and there exists some \(k\) such that \(\abs{x_n} \leq k\) for all \(n\), then we can find \(n_1 < n_2 < n_3 < n_4 < \dots\) and \(x \in \mathbb R\) such that \(x_{n_j} \to x\) as \(j \to \infty\).
 	In other words, any bounded sequence has a convergent subsequence.
@@ -161,7 +161,7 @@ \subsection{Cauchy sequences}
 	So the sequence after this point is bounded by \(1 + \abs{a_N}\).
 	The remaining terms in the sequence are only finitely many, so we can compute the maximum of all of those terms along with \(1+\abs{a_N}\) to produce a bound \(k\) for all \(n\).
 
-	By the Bolzano-Weierstrass Theorem, this sequence \(a_n\) has a convergent subsequence \(a_{n_j} \to a\).
+	By the Bolzano--Weierstrass Theorem, this sequence \(a_n\) has a convergent subsequence \(a_{n_j} \to a\).
 	We want to prove that \(a_n \to a\).
 	Given \(\varepsilon > 0\), there exists \(j_0\) such that \(\abs{a_{n_j} - a} < \varepsilon\) for all \(j \geq j_0\).
 	Also, \(\exists N(\varepsilon)\) such that \(\abs{a_m - a_n} < \varepsilon\) for all \(m, n \geq N(\varepsilon)\).

ia/analysis/06_limit_of_a_function.tex

@@ -91,7 +91,7 @@ \subsection{Bounds of a continuous function}
 \begin{proof}
 	Suppose that such a function \(f\) is not bounded.
 	Then in particular, given any integer \(n \geq 1\), there exists \(x_n \in [a, b]\) such that \(\abs{f(x_n)} > n\).
-	By the Bolzano-Weierstrass theorem, the sequence \(x_n\), which is bounded by \(a \leq x_n \leq b\), has a convergent subsequence \(x_{n_j} \to x\), such that \(x \in [a, b]\).
+	By the Bolzano--Weierstrass theorem, the sequence \(x_n\), which is bounded by \(a \leq x_n \leq b\), has a convergent subsequence \(x_{n_j} \to x\), such that \(x \in [a, b]\).
 	Then by continuity of \(f\), \(f(x_{n_j}) \to f(x)\).
 	But \(\abs{f(x_{n_j})} > n_j \to \infty\).
 	This is a contradiction.
@@ -111,7 +111,7 @@ \subsection{Bounds of a continuous function}
 	By the above theorem, \(A\) is bounded.
 	It is also non-empty, hence it has a supremum \(M = \sup A\) (and analogously an infimum \(\inf A\), whose proof is almost identical).
 	Then by the definition of the supremum, given an integer \(n \geq 1\) there exists \(x_n \in [a, b]\) such that \(M - \frac{1}{n} < f(x_n) \leq M\).
-	By the Bolzano-Weierstrass theorem, there exists a convergent subsequence \(x_{n_j} \to x \in [a, b]\).
+	By the Bolzano--Weierstrass theorem, there exists a convergent subsequence \(x_{n_j} \to x \in [a, b]\).
 	Since \(f(x_{n_j}) \to M\), then by continuity, \(f(x) = M\).
 \end{proof}
 Here is an alternative proof of the same theorem.

ia/analysis/10_integration.tex

@@ -183,7 +183,7 @@ \subsection{Monotonic and continuous functions}
 	Then there exists some \(\varepsilon > 0\) such that for all \(\delta > 0\) there exist \(x, y \in [a, b]\) such that \(\abs{x-y} < \delta\) but \(\abs{f(x) - f(y)} \geq \varepsilon\).
 	Let \(\delta = \frac{1}{n}\).
 	For this choice, we can find sequences \(x_n\) and \(y_n\) with \(\abs{x_n - y_n} < \frac{1}{n}\) but \(\abs{f(x_n) - f(y_n)} \geq \varepsilon\).
-	By the Bolzano-Weierstrass theorem, since we are working in a closed bounded interval, the \(x_n\) and \(y_n\) have convergent subsequences that tend to \(c\) and \(d\).
+	By the Bolzano--Weierstrass theorem, since we are working in a closed bounded interval, the \(x_n\) and \(y_n\) have convergent subsequences that tend to \(c\) and \(d\).
 	Then by the triangle inequality,
 	\[
 		\abs{y_{n_k} - c} \leq \abs{y_{n_k} - x_{n_k}} + \abs{x_{n_k} - c} \to 0

ia/analysis/14_uses_of_integration.tex

@@ -145,13 +145,13 @@ \subsection{Integral test for series convergence}
 	\]
 	which diverges, so by the integral test the series diverges.
 \end{example}
-\begin{corollary}[Euler-Mascheroni Constant]
+\begin{corollary}[Euler--Mascheroni Constant]
 	As \(n \to \infty\),
 	\[
 		\sum_1^n \frac{1}{n} - \int_1^n \frac{1}{n} = 1 + \frac{1}{2} + \dots + \frac{1}{n} - \log n \to \gamma
 	\]
 	where \(\gamma \in [0, 1]\).
-	This is known as the Euler-Mascheroni constant.
+	This is known as the Euler--Mascheroni constant.
 	It is unknown whether \(\gamma\) is irrational.
 \end{corollary}
 

ia/dr/14_space_time_diagrams_simultaneity_and_causality.tex

@@ -12,6 +12,7 @@ \subsection{Space-time diagrams}
 				ylabel = \(ct\),
 				xtick=\empty,
 				ytick=\empty,
+				width=6cm,
 			]
 
 			\fill (axis cs:2,1) circle[radius=2pt];
@@ -30,9 +31,10 @@ \subsection{Space-time diagrams}
 				axis lines = center,
 				xlabel = \(x\),
 				ylabel = \(ct\),
+				width=6cm,
 			]
 
-			\addplot [red!90!black, fill=red, fill opacity=0.3] coordinates {
+			\addplot [fill=black!30!white, fill opacity=0.3] coordinates {
 					(0, 0) (2, 2) (-2, 2) (2, -2) (-2, -2)
 				} -- cycle;
 		\end{axis}
@@ -53,6 +55,7 @@ \subsection{Space-time diagrams}
 				ymax=2,
 				xtick=\empty,
 				ytick=\empty,
+				width=6cm,
 			]
 
 			\draw[color=lightgray] (axis cs: -2.5, -2) -- (axis cs: -1.5, 2);
@@ -129,9 +132,9 @@ \subsection{Causality}
 				ylabel=\(y\),
 				zlabel={\(ct\)},
 				%shader=flat,
-				colormap={blue}{rgb=(0.8,0.8,1.0) rgb=(0.8,0.8,1.0)}
+				colormap={gray}{rgb=(0.9,0.9,0.9) rgb=(0.9,0.9,0.9)}
 			]
-			\addplot3 [surf, domain=-1:1, y domain=0:2*pi, z buffer=sort, colormap name=blue] ({x*cos(deg(y))},{x*sin(deg(y))},{x});
+			\addplot3 [surf, domain=-1:1, y domain=0:2*pi, z buffer=sort, colormap name=gray] ({x*cos(deg(y))},{x*sin(deg(y))},{x});
 		\end{axis}
 	\end{tikzpicture}
 \end{center}
@@ -198,7 +201,7 @@ \subsection{The twin paradox}
 To rectify this, consider the frame of reference of \(B\)'s outward journey.
 At \(E\), \(x' = 0\) and \(t' = T / \gamma\).
 Consider an event \(G\) simultaneous to \(E\) in the frame of reference of \(S'\).
-The blue line is a line of constant \(t'\).
+The new line drawn in the following diagram is a line of constant \(t'\).
 \begin{center}
 	\begin{tikzpicture}
 		\begin{axis}[
@@ -219,8 +222,8 @@ \subsection{The twin paradox}
 			\node[anchor=south] at (axis cs:1,0) {\(P\)};
 			\node[anchor=north west] at (axis cs:1,2) {\(E\)};
 			\draw[dashed,color=lightgray] (axis cs: 1, 0) -- (axis cs: 1, 5);
-			\draw[color=blue] (axis cs: 0, 1.75) -- (axis cs: 2, 2.25);
-			\node[blue,anchor=north west] at (axis cs:0,1.75) {\(G\)};
+			\draw(axis cs: 0, 1.75) -- (axis cs: 2, 2.25);
+			\node[anchor=north west] at (axis cs:0,1.75) {\(G\)};
 
 			\draw[->-=0.5] (axis cs: 0, 0) -- (axis cs: 1, 2);
 			\draw[->-=0.5] (axis cs: 1, 2) -- (axis cs: 0, 4);
@@ -253,17 +256,17 @@ \subsection{The twin paradox}
 			\node[anchor=south] at (axis cs:1,0) {\(P\)};
 			\node[anchor=west,xshift=0.4cm] at (axis cs:1,2) {\(E\)};
 			\draw[dashed,color=lightgray] (axis cs: 1, 0) -- (axis cs: 1, 5);
-			\draw[color=red] (axis cs: 0, 2.25) -- (axis cs: 2, 1.75);
-			\node[red,anchor=south west] at (axis cs:0,2.25) {\(H\)};
-			\draw[color=blue] (axis cs: 0, 1.75) -- (axis cs: 2, 2.25);
-			\node[blue,anchor=north west] at (axis cs:0,1.75) {\(G\)};
+			\draw(axis cs: 0, 2.25) -- (axis cs: 2, 1.75);
+			\node[anchor=south west] at (axis cs:0,2.25) {\(H\)};
+			\draw(axis cs: 0, 1.75) -- (axis cs: 2, 2.25);
+			\node[anchor=north west] at (axis cs:0,1.75) {\(G\)};
 
 			\draw[->-=0.5] (axis cs: 0, 0) -- (axis cs: 1, 2);
 			\draw[->-=0.5] (axis cs: 1, 2) -- (axis cs: 0, 4);
 		\end{axis}
 	\end{tikzpicture}
 \end{center}
-The red line is a line of constant \(t'\) as measured by \(B\) on the return journey, at \(E\).
+The new line is a line of constant \(t'\) as measured by \(B\) on the return journey, at \(E\).
 So for the return journey, \(A\) sees \(B\) age from the event \(E\) to the event \(F\).
 However, \(B\) sees \(A\) age from the event \(H\) to the event \(F\).
 So there is a time gap between \(G\) and \(H\) as observed by \(B\), which is not considered by the naive model of this problem.
@@ -291,8 +294,8 @@ \subsection{Length contraction}
 For observers on the platform, the train indeed contracts to length \(L\), so indeed it fits.
 On the other hand, for observers on the train, the platform contracts to a length \(\frac{1}{2}L\), so the train would not fit.
 To resolve the uncertainty, we will draw a spacetime diagram, from the frame of reference \(S\) where the platform is stationary.
-The red lines represent the end points of the platform.
-The world lines for the end points of the train are in blue.
+The vertical lines represent the end points of the platform.
+The world lines for the end points of the train are the diagonal lines intersecting \( E \) and \( F \).
 \(E\) is the event when the rear of the train is at the rear of the platform, and \(F\) is the event where the front of the train is at the front of the platform.
 \begin{center}
 	\begin{tikzpicture}
@@ -313,15 +316,15 @@ \subsection{Length contraction}
 			\draw[->,>=stealth] (axis cs: -2, -0.5) -- (axis cs: 3, 0.75);
 			\node[anchor=north east] at (axis cs:3,0.75) {\(x'\)};
 
-			\draw[color=red] (axis cs: 1, -1) -- (axis cs: 1, 3);
-			\draw[color=red] (axis cs: 2, -1) -- (axis cs: 2, 3);
+			\draw(axis cs: 1, -1) -- (axis cs: 1, 3);
+			\draw(axis cs: 2, -1) -- (axis cs: 2, 3);
 
 			\draw[dashed,color=lightgray] (axis cs: -1, 1) -- (axis cs: 3, 1);
 			\node[anchor=east] at (axis cs:1, 1) {\(E\)};
 			\node[anchor=west] at (axis cs:2, 1) {\(F\)};
 
-			\draw[color=blue] (axis cs: 0, -3) -- (axis cs: 2, 5);
-			\draw[color=blue] (axis cs: 1, -3) -- (axis cs: 3, 5);
+			\draw(axis cs: 0, -3) -- (axis cs: 2, 5);
+			\draw(axis cs: 1, -3) -- (axis cs: 3, 5);
 
 			%\draw[dashed,color=lightgray] (axis cs: 1, 0) -- (axis cs: 2, 5);
 		\end{axis}

ia/dr/import.tex

@@ -1,3 +1,11 @@
 \chapter[Dynamics and Relativity \\ \textnormal{\emph{Lectured in Lent \oldstylenums{2021} by \textsc{Prof.\ P.\ H.\ Haynes}}}]{Dynamics and Relativity}
 \emph{\Large Lectured in Lent \oldstylenums{2021} by \textsc{Prof.\ P.\ H.\ Haynes}}
+
+In the first part of this course, we study the classical laws of motion.
+We apply physical laws to study various phenomena such as gravity, friction, and orbits.
+Many such laws take the form of differential equations, and by solving these equations we can compute things like trajectories of particles.
+
+In the second part, we study special relativity.
+We explore things like time dilation and the twin paradox, and how the laws of physics seem to change when particles are travelling very close to the speed of light.
+
 \subfile{../dr/main.tex}

ib/antop/02_uniform_continuity.tex

@@ -53,7 +53,7 @@ \subsection{Properties of continuous functions}
 \begin{proof}
 	Suppose there exists \( \varepsilon > 0 \) such that \( \forall \delta > 0, \exists x,y \in [a,b], \abs{y-x} < \delta, \abs{f(y)-f(x)} \geq \varepsilon \).
 	In particular, we can construct a sequence \( (\delta_n) \) defined by \( \delta_n = \frac{1}{n} \), and we can construct sequences \( x_n, y_n \in [a,b] \) such that \( \abs{y_n-x_n} < \frac{1}{n} \) but \( \abs{f(y_n) - f(x_n)} \geq \varepsilon \).
-	By the Bolzano-Weierstrass theorem, there exists a subsequence \( (x_{k_n}) \) that converges.
+	By the Bolzano--Weierstrass theorem, there exists a subsequence \( (x_{k_n}) \) that converges.
 	Now, let \( x \) be the limit of the subsequence, \( \lim_{n \to \infty} x_{k_n} \).
 	Then \( x \in [a,b] \) since the interval is closed.
 	Then, \( \abs{y_{k_n} - x} \leq \abs{y_{k_n} - x_{k_n}} + \abs{x_{k_n} - x} < \frac{1}{n} + \abs{x_{k_n} - x} \to 0 \).

ib/antop/06_contraction_mapping_theorem.tex

@@ -96,7 +96,7 @@ \subsection{Application of contraction mapping theorem}
 	So we can combine the solutions together to yield a unique solution on \( [0,1) \).
 \end{remark}
 
-\subsection{Lindel\"of-Picard theorem}
+\subsection{Lindel\"of--Picard theorem}
 \begin{theorem}
 	Let \( n \in \mathbb N \), \( y_0 \in \mathbb R^n \), and \( a,b,R \in \mathbb R \), such that \( a < b \) and \( R > 0 \).
 	Let \( \phi \colon [a,b] \times \mathcal B_R(y_0) \to \mathbb R^n \) be a continuous function.

ib/antop/07_topology.tex

@@ -551,7 +551,7 @@ \subsection{Continuity of functions in quotient spaces}
 	Suppose \( f \) is not an open map, so there exists an open set \( U \) in \( \mathbb R \) such that \( f(U) \) is not open in \( S^1 \).
 	So \( S^1 \setminus f(U) \) is not closed, so there exists a sequence \( (z_n) \) in this complement and \( z \in f(U) \) such that \( z_n \to z \).
 	\( f \) is surjective so for all \( n \in N \) we can choose \( x_n \in [0,1] \) such that \( f(x_n) = z_n \).
-	This is a bounded sequence, so by the Bolzano-Weierstrass theorem, without loss of generality we can let \( x_n \to x \in [0,1] \).
+	This is a bounded sequence, so by the Bolzan-eierstrass theorem, without loss of generality we can let \( x_n \to x \in [0,1] \).
 	Since \( f \) is continuous, \( f(x_n) \to f(x) \), so \( z_n \to z \).
 	But since \( z_n \not\in f(U) \), we have \( x_n \in \mathbb R \setminus U \).
 	Since the complement is closed and \( x_n \to x \), we have \( x \in \mathbb R \setminus U \) so \( x \not\in U \).

ib/antop/09_compactness.tex

@@ -239,7 +239,7 @@ \subsection{Heine--Borel theorem}
 \end{proof}
 \begin{example}
 	Closed balls \( \mathcal B_r(x) \) in \( \mathbb R^n \) are compact.
-	The start of the proof for the Lindel\"of-Picard theorem now makes more sense.
+	The start of the proof for the Lindel\"of--Picard theorem now makes more sense.
 \end{example}
 
 \subsection{Sequential compactness}
@@ -249,12 +249,12 @@ \subsection{Sequential compactness}
 	Note that if \( L \subset M \subset \mathbb N \), then \( (x_n)_{n \in L} \) is a subsequence of \( (x_n)_{n \in M} \).
 \end{definition}
 \begin{example}
-	Any closed and bounded subset of \( \mathbb R \) is sequentially compact by the Bolzano-Weierstrass theorem.
+	Any closed and bounded subset of \( \mathbb R \) is sequentially compact by the Bolzano--Weierstrass theorem.
 	Similarly, any closed and bounded subset \( K \) of \( \mathbb R^n \) is sequentially compact.
 	Indeed, let \( (x_m) \) be a sequence in \( K \).
 	Then, writing \( x_m = (x_{m,1}, \dots, x_{m,n}) \), since \( K \) is bounded we have that \( (x_{m,j}) \) is bounded for all \( j \).
-	Applying the Bolzano-Weierstrass theorem to the first coordinate, we find \( M_1 \subset \mathbb N \) such that \( (x_{m,1})_{m \in M_1} \) converges in \( \mathbb R \).
-	Now, \( (x_{m,2})_{m \in M_1} \) is bounded in \( \mathbb R \), so again applying the Bolzano-Weierstrass theorem, we can find \( M_2 \subset \mathbb N \) such that \( (x_{m,2})_{m \in M_2} \) converges.
+	Applying the Bolzano--Weierstrass theorem to the first coordinate, we find \( M_1 \subset \mathbb N \) such that \( (x_{m,1})_{m \in M_1} \) converges in \( \mathbb R \).
+	Now, \( (x_{m,2})_{m \in M_1} \) is bounded in \( \mathbb R \), so again applying the Bolzano--Weierstrass theorem, we can find \( M_2 \subset \mathbb N \) such that \( (x_{m,2})_{m \in M_2} \) converges.
 	Note that \( (x_{m,1})_{m \in M_2} \) converges.
 	So inductively we can find \( M_1 \supset \dots \supset M_n \) such that \( (x_{m,j})_{m \in M_n} \) converges for all \( j \).
 	Hence \( (x_m)_{m \in M_n} \) converges in \( \mathbb R^n \).
@@ -363,7 +363,7 @@ \subsection{Compactness and sequential compactness in metric spaces}
 	But this contradicts the fact that \( \mathcal U \) does not finitely cover \( A_n \), but we have constructed a cover using just one open set.
 \end{proof}
 \begin{remark}
-	We can now deduce the one direction of the Heine--Borel theorem from the Bolzano-Weierstrass theorem; closed and bounded subsets of \( \mathbb R^n \) are compact.
+	We can now deduce the one direction of the Heine--Borel theorem from the Bolzano--Weierstrass theorem; closed and bounded subsets of \( \mathbb R^n \) are compact.
 	Similarly, we can check that the product of sequentially compact topological spaces is sequentially compact in the product topology.
 	This yields a new proof for Tychonov's theorem for metric spaces.
 	In general, there exist topological spaces that are compact but not sequentially compact, and conversely there exist topological spaces which are sequentially compact but not compact.

ib/ca/02_integration.tex

@@ -665,7 +665,7 @@ \subsection{Analytic continuation}
 	Let \( f(z) = \sum_{n=0}^\infty \exp(-2^{n/2}) z^{2^n} \), which has unit radius of convergence.
 	\( f \), and its derivatives of any order, are uniformly continuous on the closed disc \( \overline{D(0,1)} \).
 	However, we can prove that it has natural boundary \( \partial D(0,1) \), using the following theorem which will not be proven.
-	\begin{theorem}[Ostrowski-Hadamard gap theorem]
+	\begin{theorem}[Ostrowski--Hadamard gap theorem]
 		Let \( (p_n) \) be a sequence of positive integers such that \( p_{n+1} > (1+\delta)p_n \) for all \( n \) and some fixed \( \delta > 0 \).
 		If \( (c_n) \) is a sequence of complex numbers such that \( f(z) = \sum_{n=0}^\infty c_n z^{p_n} \) has unit radius of convergence, then \( \partial D(0,1) \) is the natural boundary of \( f \).
 	\end{theorem}

ib/ca/04_singularities.tex

@@ -156,7 +156,7 @@ \subsection{Essential singularities}
 	This is because \( a \) is not a pole.
 	We can generalise this further.
 \end{remark}
-\begin{theorem}[Casorati-Weierstrass theorem]
+\begin{theorem}[Casorati--Weierstrass theorem]
 	If \( f \colon U \setminus \qty{a} \to \mathbb C \) is holomorphic and \( a \in U \) is an essential singularity of \( f \), then for any \( \varepsilon > 0 \), the set \( f(D(a,\varepsilon) \setminus \qty{a}) \) is dense in \( \mathbb C \).
 \end{theorem}
 The proof is an exercise on the second example sheet.

ib/geom/04_geometry_of_surfaces.tex

@@ -691,7 +691,7 @@ \subsection{Elliptic, hyperbolic, and parabolic points}
 	We will also see (on an example sheet) that if such a local parametrisation exists, then \( \kappa \) can be expressed as a function just of \( G \).
 	This allows us to approach the proof of the \textit{theorema egregium} from a more conceptual way, since we have expressed \( \kappa \) in terms of the first fundamental form alone.
 \end{remark}
-\begin{theorem}[Gauss-Bonnet theorem]
+\begin{theorem}[Gauss--Bonnet theorem]
 	If \( \Sigma \) is a compact smooth surface in \( \mathbb R^3 \), then
 	\[
 		\int_\Sigma \kappa \dd{A_\Sigma} = 2 \pi \chi(\Sigma)