Some changes

Author: dalcde

.gitignore

@@ -15,3 +15,6 @@
 todo
 tmp/
 Makefile
+*.fdb_latexmk
+*.fls
+*.synctex.gz

IA_L/vector_calculus.tex

@@ -588,7 +588,7 @@ \subsection{Work and potential energy}
 \]
 the rate of change of energy is
 \[
-  \frac{\d}{\d t}T(t) = m\dot{\mathbf{r}}\cdot \ddot{\mathbf{r}} = \mathbf{F}\cdot \mathbf{r}.
+  \frac{\d}{\d t}T(t) = m\dot{\mathbf{r}}\cdot \ddot{\mathbf{r}} = \mathbf{F}\cdot \dot{\mathbf{r}}.
 \]
 Suppose the path of particle is a curve $C$ from $\mathbf{a} = \mathbf{r}(\alpha)$ to $\mathbf{b} = \mathbf{r}(\beta)$, Then
 \[
@@ -1250,7 +1250,7 @@ \subsection{Surface integral of vector fields}
   \]
   So $\int_S \mathbf{u}\cdot \d \mathbf{S}$ is the \emph{rate} of volume crossing $S$.
 
-  For example, let $\mathbf{u} = (-x, 0, z)$ and $S$ be the section of a sphere of radius $a$ with $0 \leq \varphi \leq$ and $0 \leq \theta \leq \alpha$. Then
+  For example, let $\mathbf{u} = (-x, 0, z)$ and $S$ be the section of a sphere of radius $a$ with $0 \leq \varphi \leq 2\pi$ and $0 \leq \theta \leq \alpha$. Then
   \[
     \d \mathbf{S} = a^2 \sin \theta \mathbf{n}\;\d \varphi \;\d \theta,
   \]

IA_M/groups.tex

@@ -5,7 +5,6 @@
 \def\nyear {2014}
 \def\nlecturer {J.\ Goedecke}
 \def\ncourse {Groups}
-\def\nofficial {http://www.dpmms.cam.ac.uk/~jg352/pdf/GroupsNotes.pdf}
 
 \input{header}
 

IA_M/vectors_and_matrices.tex

@@ -350,8 +350,8 @@ \subsubsection{Geometric picture (\texorpdfstring{$\R^2$}{R2} and \texorpdfstrin
 
 The cosine rule can be derived as follows:
 \begin{align*}
-  |\overrightarrow{BC}|^2 &= |\overrightarrow{AB} + \overrightarrow{AC}|^2\\
-  &= (\overrightarrow{AB} + \overrightarrow{AC})\cdot (\overrightarrow{AB} + \overrightarrow{AC})\\
+  |\overrightarrow{BC}|^2 &= |\overrightarrow{AC} - \overrightarrow{AB}|^2\\
+  &= (\overrightarrow{AC} - \overrightarrow{AB})\cdot (\overrightarrow{AC} - \overrightarrow{AB})\\
   &= |\overrightarrow{AB}|^2 + |\overrightarrow{AC}|^2 - 2|\overrightarrow{AB}||\overrightarrow{AC}|\cos\theta
 \end{align*}
 We will later come up with a convenient algebraic way to evaluate this scalar product.

IB_E/optimisation.tex

@@ -5,7 +5,7 @@
 \def\nyear {2015}
 \def\nlecturer {F.\ A.\ Fischer}
 \def\ncourse {Optimisation}
-\def\nofficial {http://www.statslab.cam.ac.uk/~ff271/teaching/opt/}
+\def\nofficial {http://www.maths.qmul.ac.uk/~ffischer/teaching/opt/}
 
 \input{header}
 

IB_M/linear_algebra.tex

@@ -1259,7 +1259,7 @@ \subsection{Dual space}
 \begin{proof}
   Since linear maps are characterized by their values on a basis, there exists unique choices for $\varepsilon_1, \cdots, \varepsilon_n \in V^*$. Now we show that $(\varepsilon_1, \cdots, \varepsilon_n)$ is a basis.
 
-  Suppose $\theta \in V^*$. We show that we can write it uniquely as a combination of $\varepsilon_1, \cdots, \varepsilon_n$. We have $\theta = \sum_{i = 1}^n \lambda_i \varepsilon_i$ if and only if $\theta(\mathbf{e}_j) = \sum_{i = 1}^n \varepsilon_i(\mathbf{e}_j)$ (for all $j$) if and only if $\lambda_j = \theta(\mathbf{e}_j)$. So we have uniqueness and existence.
+  Suppose $\theta \in V^*$. We show that we can write it uniquely as a combination of $\varepsilon_1, \cdots, \varepsilon_n$. We have $\theta = \sum_{i = 1}^n \lambda_i \varepsilon_i$ if and only if $\theta(\mathbf{e}_j) = \sum_{i = 1}^n \lambda_i \varepsilon_i(\mathbf{e}_j)$ (for all $j$) if and only if $\lambda_j = \theta(\mathbf{e}_j)$. So we have uniqueness and existence.
 \end{proof}
 
 \begin{cor}
@@ -1302,7 +1302,7 @@ \subsection{Dual space}
   \]
   So we can compute
   \begin{align*}
-    \left(\sum_{\ell = 1}^\infty P_{i\ell}\eta_\ell\right)(\mathbf{e}_j) &= \left(\sum_{\ell = 1}^\infty P_{i\ell}\eta_\ell\right)\left(\sum_{k = 1}^n Q_{kj}\mathbf{f}_k\right)\\
+    \left(\sum_{\ell = 1}^n P_{i\ell}\eta_\ell\right)(\mathbf{e}_j) &= \left(\sum_{\ell = 1}^n P_{i\ell}\eta_\ell\right)\left(\sum_{k = 1}^n Q_{kj}\mathbf{f}_k\right)\\
     &= \sum_{k, \ell} P_{i\ell}\delta_{\ell k} Q_{kj}\\
     &= \sum_{k, \ell} P_{i\ell} Q_{\ell j}\\
     &= [PQ]_{ij}\\

III_M/modern_statistical_methods.tex

@@ -5,7 +5,7 @@
 \def\nyear {2017}
 \def\nlecturer {R.\ D.\ Shah}
 \def\ncourse {Modern Statistical Methods}
-\def\nofficial {https://www.statslab.cam.ac.uk/~rds37/modern_statistical_methods}
+\def\nofficial {https://www.dpmms.cam.ac.uk/~rds37/modern_stat_methods.html}
 
 \input{header}
 

III_M/quantum_computation.tex

@@ -5,7 +5,7 @@
 \def\nyear {2016}
 \def\nlecturer {R.\ Jozsa}
 \def\ncourse {Quantum Computation}
-\def\nofficial {https://www.qi.damtp.cam.ac.uk/node/261}
+\def\nofficial {http://www.qi.damtp.cam.ac.uk/part-iii-quantum-computation}
 
 \input{header}