Semester 1 Exam Formula Sheets

readme.md

Exam Formula Sheets

Engineering Maths 1 Blocks 1-3

Analogue Electronics 1

ae1.pdf

ae1.tex

\documentclass[a4paper]{article}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{circuitikz}
%\usepackage{minipage}
\usepackage{graphicx} % Required for inserting images

\usepackage{fancyhdr}
\makeatletter
  \setlength{\headheight}{15pt}
  \lhead{\@author}\chead{\@title}\rhead{\today}
\makeatother
\pagestyle{fancy}

\title{AE1 Design Exam Formulas}
\author{3039284c }
\date{December 2024}

% imaginary unit
\newcommand{\iu}{{j\mkern1mu}}
\newcommand{\textreal}{$\mathbb{R}$eal\ }
% circuit algebra
%\newcommand{\parallelwith}{\mathbin{\|}}

\begin{document}

\section{Ohm's Law, KCL, KVL, Power}

Ohm's Law is
\begin{align*}
    V &= IR. \\
    &\text{ or} \\
    V &= IZ. \\
\end{align*}
Kirchhoff's Current Law says that for any node:
\begin{equation*}
    \sum I_{in} = \sum I_{out}.
\end{equation*}
Kirchhoff's Voltage Law says that for a closed loop circuit:
\begin{equation*}
    \sum V = 0.
\end{equation*}
Power is given by:
\begin{align*}
    P   &= IV \\
        &= I^2R \\
        &= \frac{V^2}{R}. \\
        &\text{ or} \\
    P   &= I|Z|.
\end{align*}
Conductance is the inverse of resistance:
\begin{equation*}
    G = R^{-1} = \frac{1}{R}.
\end{equation*}
Admittance is the inverse of impedance:
\begin{equation*}
    Y = Z^{-1} = \frac{1}{Z}.
\end{equation*}

\section{Parallel, Series, Delta-Wye}

Series:
\begin{minipage}{0.5\textwidth}
%\floatbox[{\capbeside\thisfloatsetup{capbesideposition={right,top},capbesidewidth=4cm}}]{figure}[\FBwidth]
    \centering
    \begin{circuitikz} \draw
        (0,0) to[resistor, l=$\mathrm R_1$] (2,0)
            to[resistor, l=$\mathrm R_2$] (4, 0);
    \end{circuitikz}
\end{minipage}
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{equation*}
        R_t = R_1 + R_2.
    \end{equation*}
\end{minipage}
\vspace{2em}

\noindent Parallel:
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{circuitikz} \draw
        (-1, 0) -- (0, 0)
            (0,0) -- (0, 1) 
            to[resistor, l=$\mathrm R_1$] (2, 1) -- (2, -1)
            to[resistor, l=$\mathrm R_2$] (0, -1) -- (0, 0)
        (2, 0) -- (3, 0);
    \end{circuitikz}
\end{minipage}
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{align*}
        &R_t = R_1 \parallel R_2 \\
        &G_t = G_1 + G_2. \\
        &\frac{1}{R_t} = \frac{1}{\frac{1}{R_1} + \frac{1}{R_2}}. \\
        \text{Given}&\text{ two resistors:} \\
        &R_t = \frac{R_1 \cdot R_2}{R_1 + R_2}.
    \end{align*}
\end{minipage}
\vspace{2em}

\begin{minipage}{\textwidth}
\noindent Delta-Wye:
\begin{center}
    \begin{circuitikz}[scale=0.8, transform shape]  \draw
        % Delta
        (0, 0) to [resistor, l=$\mathrm R_A$] (4.5, 8)
            to [resistor, l=$\mathrm R_B$] (9, 0)
            to [resistor, l=$\mathrm R_C$] (0, 0);
        % Wye
        \draw 
        (4.5, 6) to [resistor, l=$\mathrm R_1$] (4.5, 3.5)
            to [resistor, l=$\mathrm R_2$] (2.5, 1.5)
        (4.5, 3.5) to [resistor, l=$\mathrm R_3$] (6.5, 1.5);
    \end{circuitikz}
\end{center}
\begin{align*}
    \text{Delta} &\to \text{Wye} 
    \\
    R_1 &= R_A + R_B \parallel R_C \\
    R_1 &= \frac{R_A \cdot R_B}{R_A+R_B+R_C}.
\end{align*}
\end{minipage}

\section{Impedance}

\begin{minipage}{0.5\textwidth}
    \centering
    \begin{circuitikz} \draw
        (0,0) to[resistor] (2,0);
    \end{circuitikz}
\end{minipage}
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{equation*}
        \qquad Z = R.
    \end{equation*}
\end{minipage}
\vspace{2em}

\begin{minipage}{0.5\textwidth}
    \centering
    \begin{circuitikz} \draw
        (0,0) to[capacitor] (2,0);
    \end{circuitikz}
\end{minipage}
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{align*}
        Q &= CV. \\
        Z &= \frac{1}{\iu\omega C}.
    \end{align*}
    The current leads the voltage.
\end{minipage}
\vspace{2em}

\begin{minipage}{0.5\textwidth}
    \centering
    \begin{circuitikz} \draw
        (0,0) to[inductor] (2,0);
    \end{circuitikz}
\end{minipage}
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{equation*}
        Z = \iu \omega L.
    \end{equation*}
    The voltage leads the current.
\end{minipage}

\section{Filters}

\subsection{Low-Pass}
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{circuitikz} \draw
        (0,0) to[resistor] (4,0)
            to[capacitor] (4,-2)
            to (4, -2) node[ground]{}
        (4,0) -- (6, 0);
    \end{circuitikz}
\end{minipage}
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{align*}
        &\omega_0 = \frac{1}{RC}. \\
        &\frac{V_o}{V_i} = \frac{1}{1 + \iu\omega C R} = \frac{1}{1 + \iu\frac{\omega}{\omega_0}}. \\
        &\left|\frac{V_o}{V_i}\right| = \frac{1}{\sqrt{1 + (\omega C R)^2}} = \frac{1}{\sqrt{1 + (\frac{\omega}{\omega_0})^2}}.
    \end{align*}
\end{minipage}
\vspace{1em}
or: \\
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{circuitikz} \draw
        (0,0) to[inductor] (4,0)
            to[resistor] (4,-2)
            to (4, -2) node[ground]{}
        (4,0) -- (6, 0);
    \end{circuitikz}
\end{minipage}
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{equation*}
        \omega_0 = \frac{R}{L}.
    \end{equation*}
\end{minipage}

\subsection{High-Pass}
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{circuitikz} \draw
        (0,0) to[capacitor] (4,0)
            to[resistor] (4,-2)
            to (4, -2) node[ground]{}
        (4,0) -- (6, 0);
    \end{circuitikz}
\end{minipage}
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{align*}
        &\omega_0 = \frac{1}{RC}. \\
        &\frac{V_o}{V_i} = \frac{1}{1 + \frac{1}{\iu\omega C R}} = \frac{1}{1 + \iu\frac{\omega_0}{\omega}}. \\
        &\left|\frac{V_o}{V_i}\right| = \frac{1}{\sqrt{1 + \left(\frac{1}{\omega C R}\right)^2}} = \frac{1}{\sqrt{1 + (\frac{\omega_0}{\omega})^2}}.
    \end{align*}
\end{minipage}
\vspace{1em}
or: \\
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{circuitikz} \draw
        (0,0) to[resistor] (4,0)
            to[inductor] (4,-2)
            to (4, -2) node[ground]{}
        (4,0) -- (6, 0);
    \end{circuitikz}
\end{minipage}
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{equation*}
        \omega_0 = \frac{L}{R}.
    \end{equation*}
\end{minipage}

\subsection{Band-Pass}
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{circuitikz} \draw
        (0,0) to[capacitor] (2,0)
            to[inductor] (3, 0) -- (4, 0)
            to[resistor] (4,-2)
            to (4, -2) node[ground]{}
        (4,0) -- (6, 0);
    \end{circuitikz}
\end{minipage}
\begin{minipage}{0.5\textwidth}
    \centering
    \begin{equation*}
        \omega_0^2 = \frac{1}{LC}.
    \end{equation*}
    %\begin{equation*}
    %    \frac{V_o}{V_i} = \frac{1}{1 + \frac{1}{\iu\omega C R}} = \frac{1}{1 + \iu\frac{\omega_0}{\omega}}.
    %\end{equation*}
\end{minipage}
\vspace{1em}

\section{Operational Amplifiers ``Op-Amps''}

\subsection{Non-inverting configuration}
\begin{minipage}{0.6\textwidth}
    \begin{circuitikz}\draw
        (0, 0) node[op amp] (opamp) {}
        (opamp.-) to[resistor, label=$\mathrm R_i$] (-3, 0.5) -- (-4, 0.5)
            to ++(0,-3) node[ground]{}
        (opamp.+) -- ++(-1, 0) to[sinusoidal voltage source] ++(0, -2) to node[ground]{} ++(0, 0)
        (opamp.-) to[short,*-] ++(0,1.5) coordinate (leftR)
            to[resistor, label=$\mathrm R_f$] (leftR -| opamp.out)
            to[short,-*] (opamp.out)
        (opamp.out) to ++(1, 0) node[label={above:$V_o$}]{};
    \end{circuitikz}
\end{minipage}
\hfill\vline\hfill
\begin{minipage}{0.3\textwidth}
    \centering
    \begin{circuitikz} \draw
    (0, 0) node[label=$V_o$]{} 
        to [resistor, *-*, label=$\mathrm R_f$] ++(0, -2) node[label={right:$V_-=V_+=V_i$}]{}
        to [resistor, *-*, label=$\mathrm R_i$] ++(0, -2) node[label={right:0V}]{}
        to node[ground]{} ++(0,0);
    \end{circuitikz}
\end{minipage}
\vspace{2em}

For a non-inverting amplifier:
\begin{align*}
    I = \frac{V_o - V_i}{R_f} &= \frac{V_i - 0}{R_i} \\
    \therefore \frac{V_o}{V_i} &= 1 + \frac{R_f}{R_i}.
\end{align*}

\subsection{Inverting configuration}
\begin{minipage}{0.6\textwidth}
    \begin{circuitikz}\draw
        (0, 0) node[op amp] (opamp) {}
        (opamp.-) to[resistor, label=$\mathrm R_i$] ++(-2, 0) -- ++(-1, 0)
            to[sinusoidal voltage source, label=$V_i$] ++(0, -2) to ++(0, 0) node[ground]{}
        (opamp.+) -- ++(-1, 0) to ++(0, -1) node[ground]{}
        (opamp.-) to[short,*-] ++(0,1.5) coordinate (leftR)
            to[resistor, label=$\mathrm R_f$] (leftR -| opamp.out)
            to[short,-*] (opamp.out)
        (opamp.out) to ++(1, 0) node[label={above:$V_o$}]{};
    \end{circuitikz}
\end{minipage}
\hfill\vline\hfill
\begin{minipage}{0.3\textwidth}
    \centering
    \begin{circuitikz} \draw
    (0, 0) node[label=$V_o$]{} 
        to [resistor, *-*, label=$\mathrm R_f$] ++(0, -2) node[label={right:$V_-=V_+=0\mathrm V$}]{}
        to [resistor, *-*, label=$\mathrm R_i$] ++(0, -2) node[label={right:$V_i$}]{}
        to node[ground]{} ++(0,0);
    \end{circuitikz}
\end{minipage}
\vspace{2em}

\noindent For an inverting amplifier, consider the current:
\begin{align*}
    I = \frac{V_i - V_o}{R_f+R_i} =  \frac{Vi - 0}{R_i} &= \frac{0 - V_o}{R_f} \\
    \therefore - \frac{V_o}{V_i} &= \frac{R_f}{R_i}.
\end{align*}

Remember that: \emph{No current flows into the terminals}; and \emph{The differential input voltage is 0}. It is obvious that $V_o$ must be negative.

\section{Miscellaneous}

An example of an Argand Diagram showing Voltage, Current and Impedance:
\begin{center}
    \includegraphics[width=0.6\textwidth]{example argand.png}
\end{center}
\vspace{2em}

Q and I in a capacitor against time:
\begin{center}
    \includegraphics[width=0.5\textwidth]{Q,I,t.png}
\end{center}
\vspace{4em}

\begin{minipage}{0.45\textwidth}
    In a capacitor the current leads the voltage.
    \begin{center}
        \includegraphics[width=\textwidth]{IleadsV.png}
    \end{center}
\end{minipage}
\hfill\vline\hfill
\begin{minipage}{0.45\textwidth}
    In an inductor the voltage leads the current.
    \begin{center}
        \includegraphics[width=\textwidth]{VleadsI.png}
    \end{center}
\end{minipage}
\vspace{2em}

\begin{minipage}{\textwidth}
Decibels are given by:
\begin{align*}
    \text{dB:}\quad         &10 \log_{10}\left(\frac{P_2}{P_1}\right) \\
    P = \frac{V^2}{R}\to    &10\log_{10}\left(\frac{V_2^2}{V_1^1}\right) \\
                           =&20\log_{10}\left(\frac{V_2}{V_1}\right)
\end{align*}
\end{minipage}

\end{document}

engmaths1_blocks1-3.pdf

engmaths1_blocks1-3.tex

\documentclass[a4paper,11pt]{article}

\title{\Large Engineering Maths 1 \\
\normalsize Formula Sheet \\
Blocks 1-3 }
\author{
Emil Carr-Ross \\
3039284C}
\date{\today}

\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{hyperref}
\usepackage{makeidx}
\usepackage[range-phrase=-,range-units=single,parse-numbers=false]{siunitx}
\usepackage{tikz}
%todo remove this
%\usepackage{todonotes}
\usepackage{titlesec}

\newcommand{\todo}[1]{}

% new page for each section
\AddToHook{cmd/section/before}{\clearpage}

% differential shorthand
\newcommand{\dydx}{\frac{\mathrm{d}y}{\mathrm{d}x}}
% matrix operators
\DeclareMathOperator{\trace}{Tr}
\newcommand{\Tr}{\trace}
\DeclareMathOperator{\image}{Im}
\DeclareMathOperator{\adj}{Adj}
\DeclareMathOperator{\eig}{eig}
%\newcommand{\Im}{\image}
% proper inline lim
\newcommand{\Lim}[1]{\raisebox{0.5ex}{\scalebox{0.8}{$\displaystyle \lim_{#1}\;$}}}
% imaginary unit
\newcommand{\iu}{{j\mkern1mu}}
\newcommand{\textreal}{$\mathbb{R}$eal\ }
% function: and :=
%\DeclareMathSymbol{:}{{\mathord\,}}

%add all emphasised words to the index
\let\oldemph\emph
\renewcommand{\emph}[1]{\index{#1}\oldemph{#1}}

\makeindex
\begin{document}

\setcounter{page}{0}
\pagenumbering{gobble}
\maketitle

\clearpage
\pagenumbering{roman}
\tableofcontents

\setcounter{page}{0}
\section{Block 1 - Numbers, Algebra, Geometry and Functions}
\pagenumbering{arabic}

    \subsection{Error}

    ``The distance from the truth''

    \emph{Absolute Error} is given by: 

    $a = a_T + \epsilon_a.$

    Absolute error is Additive.

    $\epsilon_{a+b} = \epsilon_a + \epsilon_b.$

    \emph{Relative Error} is given by:

    $r_a = \frac{\epsilon_a}{a_T} \approx \frac{\epsilon_a}{a}.$

    Relative Error is Multiplicative

    $r_{a\cdot b} = r_a + r_b.$ \\

    \emph{Linear Regression} or ``\emph{Linear Least Squares}'' minimises the greatest error between all points and a line of best fit.

    \begin{equation}
        m = \sum_{i=1}^n \frac{(x_i-\bar{x})(y_i-\bar{y})}{(x_i-\bar{x})^2}.
    \end{equation}

    \subsection{Sets}
    A \emph{set} is a collection of numbers:
    \begin{equation}
        A = \left\{ 1, 2, 3 \right\}.
    \end{equation}
    Sets can contain sets:
    \begin{equation}
        C = \left\{A, B, \{7, 8, 9\}\right\}.
    \end{equation}
    The \emph{Difference}, \emph{Union} and \emph{Intersection} set operations are notated by:
    \begin{align}
        A - B       &:= A \setminus B = \left\{ x \mid x \in A \text{ and } x \notin B\right\} & \text{Difference or Minus.} \\
        A \cup B    &:= \left\{x | x \in A \text{ or } x \in B \right\} & \text{``A cup B'' := Union.} \\
        A \cap B    &:= \left\{x | x \in A \text{ and } x \in B \right\} & \text{``A cap B'' := Intersection.}
    \end{align}
    Some common sets are:
    \begin{align}
        \mathbb{N} &:= \left\{ 1, 2, 3, \ldots\right\}                      & \text{The Natural numbers.} \\
        \mathbb{Z} &:= \left\{ \ldots, -2, -1, 0, 1, 2, \ldots \right\}     & \text{The Integers.} \\
        \mathbb{Q} &:= \left\{ \frac{p}{q} \mid p \in \mathbb{Z} \text{ and } q \in \mathbb{Z}\right\} \ & \text{The Rational numbers.} \\
        \mathbb{R} &:= \text{ ``all points on the real number line'' }      & \text{The Real numbers.}
    \end{align}
    For example, the \emph{irrational} constant $\pi \in \mathbb{R}$ ``Pi is in the Real numbers'', but $\pi \notin \mathbb{Q}$ ``Pi is not in the Rational numbers''. \\

    An \emph{interval} is a set containing all \textreal numbers between $a$ and $b$:
    \begin{align}
        \left[a, b\right]   &:=   \left\{x \in \mathbb{R} \mid a \leq x \leq b \right\}     & \text{``Closed'' Interval}. \\
        \left(a, b\right)   &:=   \left\{x \in \mathbb{R} \mid a < x < b \right\}           & \text{``Open'' Interval}. \\
        \left[a, b\right)   &:=   \left\{x \in \mathbb{R} \mid a \leq x < b \right\}        & \text{``Semi-open'' Interval}.
    \end{align}
    Note: ``Closed'' because $x$ may reach $a$ or $b$ and stop. ``Open'' because $x$ can keep getting smaller or larger but never reach $a$ or $b$.

    \subsection{Domain and Co-domain}
    
    A \emph{function} maps a set of numbers to another, and is notated by:
    \begin{align}
        f{:\,} \mathbb{X}   &\to \mathbb{Y} \\
        x               &\mapsto f(x). \nonumber
    \end{align}
    Where $\mathbb{X}$ is the \emph{domain} and $\mathbb{Y}$ is the \emph{codomain}. \\
    The \emph{Image} of such a function is all of its possible outputs:
    \begin{equation}
        \image(f) := \left\{ y \in \mathbb{Y} \mid y=f(x), x \in \mathbb{X} \right\}.
    \end{equation}

    A function may only have one mapping in the codomain for every element in the domain. This is the \emph{vertical line test}.
    
    A function is \emph{injective} if $f(x_1) = f(x_2)$, then $x_1 = x_2$, ie the \emph{horizontal line test}.

    A function is \emph{surjective} if its Image equals its codomain - ie every element in the codomain has a mapping from the domain.

    A function is \emph{periodic} if for a constant $c$, $f(x) = f(x+c)$.

    \subsection{Co-ordinate systems}
    
    \emph{Cartesian} co-ordinates are given by:
    \begin{equation}
        y = f(x).
    \end{equation}

    A function may also map to a set of n-dimensional points, $\mathbb{R}^n$.
    \begin{align}
        f{:\,} \mathbb{R}   &\to \mathbb{R}^2 \nonumber \\
        t                   &\mapsto f(t) \nonumber \\
        f(t)                &= (\ldots,\ \ldots).
    \end{align}
    Where $\mathbb{R}^n$ is defined by:
    \begin{align}
        \text{coordinates:} & \left\{ (x_1, x_2, \ldots, x_n) \mid x_1, x_2, \ldots, x_n \in \mathbb{R}\right\} \nonumber \\
        \text{origin:}      &\ (0, 0, \ldots, 0).
    \end{align}
 
    \emph{Polar co-ordinates} are defined by:
    \begin{align}
        \text{co-ordinates: }   & \left\{(r, \theta) \mid r \in \mathbb{R}^+, \theta \in [0, 2\pi) \right\} \nonumber \\
        \text{origin: }         &\  (0, 0)
    \end{align}
    where $r$ is the distance from the origin (note $\mathbb{R}^+$ - positive real numbers), and $\theta$ is the angle with the positive x-axis. \\

    Functions in one co-ordinate system can be \emph{re-parameterised} in another. \\

    A \emph{Circle} is given in Cartesian co-ordinates by:
    \begin{equation}
        (y - a)^2 + (x - b)^2 = r^2
    \end{equation}
    and may be re-parameterised in $\mathbb{R}^2$ by:
    \begin{align}
        g{:\,} [0, 2\pi)    &\to \mathbb{R}^2 \nonumber \\
        t                   &\mapsto g(t) \nonumber \\ 
        g(t)                &= \left(r\cos(t)+a, -r\sin(t)+b\right)
    \end{align}
    where the centre of the Circle is $C = (a, b)$ with radius $R$. \\

    A circle in polar-coordinates has a constant radius $r$. \\
    A \emph{straight line} in polar-coordinates has a constant angle $\theta$.


    \subsection{Polynomials}

    A polynomial is given by:

    \begin{eqnarray}
        f(x) = \sum_{i=0}^n a_ix^i.
        \\
        \deg(f) = n.
    \end{eqnarray} \\
    A polynomial $f(x)$ is \emph{monic} when $a_n=1$. \\
    A \emph{linear} polynomial has $\deg(f) = 1$. \\
    A \emph{quadratic} or \emph{binomial} polynomial has $\deg(f) = 2$ \\
    \\

    \begin{minipage}{\textwidth}
        For a linear polynomial $y = mx + c$, a perpendicular linear polynomial is given by:~    
        \begin{equation}
            y = \frac{1}{-m}x + c.
        \end{equation}
    \end{minipage} \\
    \\
    
    The \emph{choose function} gives the $i$th element from the $n$th row of Pascal's triangle, or ``How many ways can I pick $i$ objects from $n$?''.
    \begin{equation}
        {n \choose i} = \frac{n!}{(n-i)!i!}.
    \end{equation}
    This gives the coefficient of $a^{n-i}b$ in $(a-b)^i$. \\
    The binomial expansion is thus:
    \begin{equation}
        (a + b)^n \sum_{i=0}^n {n \choose i}a^{n-i}b^i.
    \end{equation} \\

    The \emph{discriminant} of a quadratic is given by:
    \begin{equation}
        \Delta = b^2 - 4ac.
    \end{equation}
    When $\Delta > 0$, the equation has two $\mathbb{R}$eal roots. \\
    When $\Delta = 0$, the equation has two $\mathbb{R}$eal and equal roots. \\
    When $\Delta < 0$, the equation has no $\mathbb{R}$eal roots. \\
    A quadratic is \emph{irreducable} when $\Delta = 0$ \\

    The roots of a quadratic are given by:
    \begin{align}
        x &= \frac{-b \pm \sqrt{\Delta}}{2a} \nonumber \\
         &= \frac{-b \pm \sqrt{b^2-4ac}}{2a}.
    \end{align}

    All polynomials can be reduced into monic: linear factors, or irreducable factors
    \begin{equation}
        p(x) = \prod_{i=0}^m (x^2 +a_i + b_i) \cdot \prod_{j=2m}^n (x-r_j).
    \end{equation}
    Left: Irreducable quadratics Right: linear factors. \\
    $r_j$ is A \emph{root} of p. \\
    \\

    A \emph{rational} function is described by:
    \begin{equation}
        f(x) = \frac{p(x)}{q(x)} \qquad p(x) \neq 0
    \end{equation}
    where $q(x)$ is the \emph{quotient}. \\
    \\
    A rational function is \emph{Strictly Proper} when $\deg(p) \leq \deg(q)$, \\
    \emph{Proper} when $\deg{p} < \deg{q}$, \\
    \emph{Improper} when $\deg{p} > \deg{q}$.

    Every strictly proper function can be written as a sum of rational functions where the quotient is a linear polynomial or an irreducable quadratic.
    \begin{equation}
        f(x) = \frac{p(x)}{q(x)} = \frac{A}{x-r_1} + \frac{B}{x-r_2} + \cdots.
    \end{equation} \\

    Rational functions may be graphed by finding the \emph{asymptotes}, maximum and minimum. 

    A rational function $f(x) = \frac{p(x)}{q(x)}$ has asymptotes as $q(x) \to 0, f(x) \to \infty$ and $p(x) \to \infty, f(x) \to \infty$.

    A rational function will have a \emph{slant asymptote} when ${\deg(p) = \deg(q)+1}$. The equation of the slant asymptote can be found by long division.

    The maximum and minimum are found when $f'(x) = 0$ (See Section~\ref{differentiation}).

    \subsection{Exponential and Logarithmic Functions}

    An exponential function is given by:
    \begin{equation}
        f(x) = x^n.
    \end{equation}
    The \emph{standard exponential function} is:
    \begin{equation}
        \exp(x) = \mathrm{e}^x
    \end{equation}
    where $\mathrm{e}$ is \emph{Euler's number}.

    A \emph{logarithm} is defined by:
    \begin{equation}
        \log_b(x) = \log_b(b^n) = n.
    \end{equation}
    ``To what power do I need to raise $b$ by such that it equals $x$?''. \\
    \\
    The \emph{natural logarithm} is $\log$ base $\mathrm{e}$:
    \begin{equation}
        \ln x = \log_\mathrm{e} x.
    \end{equation}

    \subsection{Properties of Logs}

    \begin{eqnarray}
        \log(a) + \log(b) = \log(ab).
        \\
        \log(a^n) = n\log(a).
        \\
        \log_b(x) = \frac{\ln(x)}{\ln(b)}.
    \end{eqnarray}

    \subsection{Trigonometric and Hyperbolic Trigonometric functions}

    The \emph{hyperbolic trig} functions are given by:
    \begin{align}
        \sinh(x)    &= \frac{e^x-e^{-x}}{2}. \\
        \cosh(x)    &= \frac{e^x+e^{-x}}{2}. \\
        \tanh(x)    &= \frac{\sinh(x)}{\cosh{x}} \\
                    &= \frac{e^x - e^{-x}}{e^x + e^{-x}}. \nonumber
    \end{align}

    $\sec$ (secant), $\csc$ (cosecant) and $\cot$ (cotangent) are defined by:
    \begin{eqnarray}
        \sec(x) = \frac{1}{\cos(x)}.
        \\
        \csc(x) = \frac{1}{\sin{x}}.
        \\
        \cot{x} = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)}.
    \end{eqnarray}

    The most useful circular trigonometric identities are:
    \begin{eqnarray}
        \tan x = \frac{\sin x}{\cos x}.
        \\
        \sin^2(x) + \cos^2(x) = 1.
        \\
        \sin(x+y) = \sin(x)\cos(y)+\cos(x)\sin(y) \label{sinx+y}.
        \\
        \cos(x+y) = \cos(x)\cos(y)-\sin(x)\sin(y) \label{cosx+y}.
        \\
    \end{eqnarray}
    The \emph{double} and \emph{triple} identities can be found from~\eqref{sinx+y} and~\eqref{cosx+y}, giving:
    \begin{eqnarray}
        \sin(2x) = 2\sin(x)\cos(x).
        \\
        \cos(2x) = \cos^2(x)-\sin^2(x).
        \\
        \sin(3x) = 3\sin(x)-4\sin^2(x).
        \\
        \cos(3x) = 4\cos^3(x)-3\cos(x).
    \end{eqnarray}
    The \emph{half-angle} identities are:
    \begin{eqnarray}
        \pm\cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1+\cos(x)}{2}}.
        \\
        \pm\sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1-\cos(x)}{2}}.
    \end{eqnarray}

    \subsection{Simple differentiation} \label{differentiation}

    For a function $f(x)=y=ax^n$, the derivative is given by:
    \begin{equation}
        f^\prime(x) = \dydx \cdot (ax^n) = anx^{n-1}.
    \end{equation}

    The Quotient Rule is:
    \begin{align}
        f(x)    &= \frac{p(x)}{q(x)} \nonumber \\
        f^\prime(x)   &= \frac{p^\prime(x)q(x)-p(x)q^\prime(x)}{q^2(x)}.
    \end{align}

    The derivatives of the standard exponential functions are:
    \begin{align}
        & \dydx e^x = e^x. \\
        & \dydx e^{-x} = -e^{-x}. \\
        & \dydx \ln(x) = \frac{1}{x} \qquad x > 0.
    \end{align}

    The derivatives of the circular functions are:
    \begin{align}
        \dydx \sin(x)   &= \cos(x). \\
        \dydx \cos(x)   &= -\sin(x). \\
    \end{align}
    The derivative of $\tan(x)$ is found by the quotient rule:
    \begin{align}
        \dydx \tan(x)   &= \dydx \frac{\sin(x)}{\cos(x)} \nonumber \\
                        &= \frac{\cos(x)\cos(x) + \sin(x)\sin(x)}{\cos^2(x)} \nonumber \\
                        &= \frac{1}{\cos^2(x)}.
    \end{align}

\section{Block 2 - Complex Numbers and Vector Algebra}

    \subsection{Complex Numbers}

    The \emph{imaginary unit} $\iu$ is defined as $\iu^2 = -1$.

    A \emph{complex number} $Z$ is a number which has a \emph{real} and \emph{imaginary} part.
    \begin{equation}
        Z = x + \iu y.
    \end{equation}

    Complex numbers can be plotted on the \emph{Complex Plane} or \emph{Argand Diagram}, as in Figure~\ref{argand}:

\begin{figure}[!ht]
\begin{center}
\begin{tikzpicture}
    \begin{scope}[thick,font=\scriptsize]
    
    
    \draw [->] (-4,0) -- (4,0) node [above left]  {$\mathbb{R}(z)$};
    \draw [->] (0,-4) -- (0,4) node [below right] {$\iu(z)$};

    
    \foreach \n in {-3,...,-1,1,2,...,3}{%
        \draw (\n,-3pt) -- (\n,3pt)   node [above] {$\n$};
        \draw (-3pt,\n) -- (3pt,\n)   node [right] {$\n i$};}
        
    \draw [thick, color=red] (0,0) -- (2,3);
    \draw [color=blue, fill=blue] (2,3) circle(0.05);
    \node [color=black] at (3,3) {$2 + \iu3$};
\end{scope}
\end{tikzpicture}
\end{center}
\caption{The Complex Number $2 + \iu3$ plotted on an Argand Diagram (Introduction to Complex Numbers, Ed Fowler, 2013)}
\label{argand}
\end{figure}

    The \emph{Complex Conjugate} $Z*$ of a complex number $Z  = x + \iu y$ mirrors $Z$ across the \textreal number line, and is given by $Z* = x - jy$.
    We find that:
    \begin{equation}
        ZZ* = x^2 + y^2.
    \end{equation}
    Complex quadratic roots are conjugates of each-other. \\

    Multiplication and division of complex numbers is performed by:
    \begin{align}
        Z_1Z_2  &= (x_1 + \iu y_1)(x_2 + \iu y_2) \nonumber \\ 
                &= x_1 x_2 + \iu x_1 y_2 + \iu y_1 x_2 + \iu^2 y_1 y_2 \nonumber \\
                &= x_1 x_2 + \iu x_1 y_2 + \iu y_1 x_2 - y_1 y_2.
    \end{align}
    \begin{align}
        \frac{Z_1}{Z_2} &= \frac{Z_1}{Z_2} \cdot \frac{Z_2*}{Z_2*} \nonumber \\
                        &= \frac{x_1 + \iu y_1}{x_2 + \iu y_2} \cdot \frac{x_2 - \iu y_2}{x_2 - \iu y_2} \nonumber \\
                        &= \frac{x_1 x_2 -\iu x_1 y_2 + \iu y_1 x_2 - \iu^2 y_1 y_2}{x_2^2 + y_2^2} \nonumber \\
                        &= \frac{x_1 x_2 + y_1 y_2 -\iu x_1 y_2 + \iu y_1 x_2}{x_2^2 + y_2^2}.
    \end{align}

    The \emph{modulus} of a complex number is the magnitude of the red line shown on the Argand diagram. For a complex number $Z = x + \iu y$, it is given by the Pythagorean theorem:
    \begin{equation}
        |Z| = \sqrt{x^2 + y^2}.
    \end{equation}
    The modulus has the following properites:
    \begin{align}
        ZZ* &= |Z|. \\
        |Z_1Z_2| &= |Z_1||Z_2|. \\
        \left|\frac{Z_1}{Z_2}\right| &= \frac{|Z_1|}{|Z_2|}.
    \end{align} \\

    The \emph{argument} of a complex number is the angle it makes with the positive-x axis. It is given by:
    \begin{equation}
        \arg(Z) = 
        \begin{cases}
            \arctan\left(\frac{y}{x}\right)         & \text{if } x > 0, \\
            \arctan\left(\frac{y}{x}\right) + \pi   & \text{if } x < 0, y \geq 0 \\
            \arctan\left(\frac{y}{x}\right) - \pi   & \text{if } x < 0, y < 0 \\
            +\frac{\pi}{2}                          & \text{if } x = 0, y > 0 \\
            -\frac{\pi}{2}                          & \text{if } x = 0, y < 0 \\
            \text{undefined}                        & \text{if } x = 0, y = 0
        \end{cases} \qquad -\pi < \arg(z) < \pi.
    \end{equation}
    This is known as the \emph{atan2} function in computing science.
    A more intuitive method involves calculating the angle against either the positive or negative-x axes, and adding or subtracting $\pi$ as necessary. \\
    
    A complex number can thus be represented in \emph{polar form}, where $\theta = \arg(Z)$:
    \begin{equation}
        Z = |Z|(\sin\theta + \iu\cos\theta). 
    \end{equation}
    Or in \emph{phasor} form:
    \begin{equation*}
        Z = r \angle \theta.
    \end{equation*}
    Here $r$ is the \emph{radius}, equal to the modulus, as if it were the radius of a circle traced around the Argand diagram for all values of $\theta$.

    By use of trigonometric identities, we find that:
    \begin{eqnarray}
        Z_1Z_2 = r_1r_2\left[\cos(\theta_1+\theta_2) + \iu \sin(\theta_1+\theta_2)\right]. \\
        \frac{Z_1}{Z_2} = \frac{r_1}{r_2}\left[\cos(\theta_1-\theta_2) + \iu \sin(\theta_1-\theta_2)\right].
    \end{eqnarray}
    Addition adds the argument, and division subtracts. \\
    \\

    \emph{Euler's Formula} says that:
    \begin{equation}
        re^{\iu \theta} = r(\cos\theta + \iu \sin\theta).
    \end{equation}
    From which we get \emph{Euler's Identity}, ``The most beautiful formula in mathematics'', connecting constants $e, \pi, \iu, -1$ and $0$:
    \begin{equation}
        e^{\iu\pi}-1 = 0.
    \end{equation}
    Expressing a complex number in this way is called the \emph{exponential form}. \\

    Complex numbers in polar and exponential form thus give us a definition for the circular functions:
    \begin{align}
        & e^{\iu\theta} = \cos\theta \iu \sin\theta. \nonumber \\
        & e^{-\iu\theta} = \cos\theta - \iu \sin\theta. \nonumber \\
        \therefore&~\cos\theta = \frac{e^{\iu\theta} + e^{-\iu\theta}}{2} \qquad \text{by addition.} \\
        \therefore&~\sin\theta = \frac{e^{\iu\theta} - e^{-\iu\theta}}{2} \qquad \text{by subtraction.}
    \end{align}
    And a link to the hyperbolic functions:
    \begin{align}
        & \cosh \iu x = \cos x. \\
        & \sinh \iu x = \iu \sin x. \\
        & \tanh \iu x = \iu \tan x.
    \end{align} \\
    \\

    Powers of a complex number can be taken as such:
    \begin{align}
        Z^n &= |Z|^n e^{\iu\theta n} \\
            &= r^n\left[\cos(n\theta) + \iu \sin(n\theta)\right].
    \end{align}
    Taking a power multiplies the argument. \\

    As taking a power thus rotates around the complex plane, the $n$th root must have $n$ answers. \emph{DeMoivre's Theorem} tells us that:
    \begin{equation}
        Z^{\frac{1}{n}} = r^{\frac{1}{n}}e^{\iu\left(\frac{\theta}{n} + \frac{2\pi k}{n}\right)} \qquad k \in (0,n] \cap \mathbb{Z}.
    \end{equation} \\

    The natural log of a complex number is found by:
    \begin{align}
        Z       &= |Z| e^{\iu\theta} \\
        \ln Z   &= \ln (|Z| e^{\iu\theta}) \\
                &= \ln |Z| + \ln e^{\iu\theta} \\
                &= \ln |Z| + \iu \theta.
    \end{align}

    \todo{todo: loci}

    \subsection{Vector Algebra}

\section{Block 3 - Matrix Algebra, Sequences, Series and Limits}

    \subsection{Matrices}

    A \emph{Matrix} is a rectangular array of elements. For an n\texttimes m matrix $\mathbf{A}$:
    \begin{equation}
        \mathbf{A} = 
        \begin{bmatrix}
            a_{11}  & a_{12}    & \cdots    & a_{1m} \\
            a_{21}  & \ddots    & \ddots     & \vdots \\
            \vdots  & \ddots    & \ddots    & \vdots \\
            a_{n1}  & \cdots    & \cdots    & a_{nm}
        \end{bmatrix}
    \end{equation}

    The \emph{transpose} operator $\mathbf{A}^\intercal$ interchanges the rows for columns in a matrix.\\
    A matrix is said to be \emph{symmetric} if ${\mathbf{A}^\intercal = \mathbf{A}}$ and \emph{skew-symmetrix} if ${\mathbf{A}^\intercal = -\mathbf{A}}$.
\todo{transposed matrix elements with subscripts?}

    The \emph{trace} of a square matrix is the sum of the elements along its diagonal:
    \begin{equation}
        \trace(\mathbf{A}) = \sum_{i=0}^n a_{ii}.
    \end{equation}

    \emph{Matrix Multiplication} is performed \textbf{row} by \textbf{column}. For a matrix ${\mathbf{C} = \mathbf{AB}}$:
    \begin{equation}
        c_{ij} = \sum_{k=1}^p a_{ij}b_{kj} \qquad \text{for } i = 1,\ldots,m~\text{ and }~j=1,\ldots,n.
    \end{equation}

    The \emph{Identity} or \emph{Unit} matrix $\mathbf{I}_n$ fills the diagonal of an n\texttimes n matrix, for which $a_{11},a_{22},\ldots,a_{nn}=1$.
    \begin{equation}
        \mathbf{I} =
        \begin{bmatrix}
            1 & 0 & \cdots & 0 \\
            0 & 1 & \cdots & 0 \\
            \vdots & \vdots & \ddots & \vdots \\
            0 & 0 & \cdots & 1
        \end{bmatrix}
    \end{equation}

    Multiplying by the identity matrix leaves the matrix unchanged: $\mathbf{AI}~=~\mathbf{A}$. \\
    The \emph{Image} of a matrix is given by $\image(\mathbf{A}) = \mathbf{AI}_n = \mathbf{A}$. \\

    The \emph{determinant} of a 2\texttimes2 matrix $\mathbf{A}$ is given by:
    \begin{equation}
        \det(A) = \left|\mathbf{A}\right| = a_{11}a_{22}-a_{12}a_{21}.
    \end{equation}

    A \emph{Minor} $M_{ij}$ of a matrix $\mathbf{A}$ is the determinant of $\mathbf{A}$ having deleted the $i$th row and $j$th column. \\
    A \emph{Cofactor} is a \emph{signed minor}, given by $C_{ij} = (-1)^{i+j}M_{ij}$… The sign can be found quickly by alternating pluses (+) and minuses (-1): \\
    \begin{equation*}
        \begin{bmatrix}
            + & - & + & \cdots \\
            - & + & - & \cdots \\
            + & - & + & \cdots \\
            \vdots & \vdots & \vdots & 
        \end{bmatrix}
    \end{equation*}

    The determinant of a matrix of order greater than 2\texttimes2 is evaluated by the sum of cofactors along any row $i$:
    \begin{equation}
        \det(\mathbf{A}) = \left|\mathbf{A}\right| = \sum_{j=1}^n a_{ij}C_{ij}.
    \end{equation}

    The \emph{cofactor matrix} $\mathbf{C}$ contains \textbf{all} of the cofactors of $\mathbf{A}$. \\
    \\
    The \emph{Adjoint} or \emph{Adjucate} matrix is the cofactor matrix transposed:
    \begin{equation}
        \adj(\mathbf{A}) = \mathbf{C}^\intercal.
    \end{equation}

    A Matrix is said to be \emph{singular} if $\det(\mathbf{A}) = 0$, \\
    \indent\indent or \emph{non-singular} if $\det(\mathbf{A}) \neq 0$. \\
    \\

    The \emph{inverse} of a non-singular matrix is given by:
    \begin{equation}
        \mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \adj(\mathbf{A}) \qquad \det(\mathbf{A}) \neq 0 \text{ ie } \mathbf{A} \text{ is non-singular.}
    \end{equation}
    In a 2\texttimes2 matrix this is trivial:
    \begin{align*}
        & \mathbf{A} =
        \begin{bmatrix}
            a & b \\
            c & d
        \end{bmatrix}
        \\
        & \mathbf{A}^{-1} = 
        \begin{bmatrix}
            d & -b \\
            -c & a 
        \end{bmatrix}.
    \end{align*}

    A Matrix multiplied by its inverse equals the identity matrix:
    \begin{equation}
        \mathbf{AA}^{-1} = \mathbf{I}.
    \end{equation}

    A system of linear equations can be represented by a matrix:
    \begin{align*}
        a_{11}&x_1 + a_{12}x_2 + \cdots &= b_1 \\
        a_{21}&x_1 + a_{22}x_2 + \cdots &= b_2 \\
              & \vdots                  &=~~\vdots
    \end{align*}
    \begin{equation}
        \mathbf{AX} = \mathbf{B}.
    \end{equation}
\todo{augmented matrix form}
    The solution for any system of linear equations are found by multiplying by the inverse:
    \begin{equation}
        \mathbf{I} = \mathbf{A}^{-1}\mathbf{B}.
    \end{equation}
    The \emph{trivial solution} $\mathbf{X}=0$ is when $\mathbf{A}^{-1}$ exists and $\mathbf{B}=0$.\\
    \\
\todo{todo: row-echelon form, augmented matrix form.}
    The \emph{Rank} of a matrix is the number of linearly independent variables. Informally, it is the number of non-zero rows in \emph{Row-Echelon Form}. An nxn matrix is said to be \emph{full rank} when $r(\mathbf{A}) = n$. \\
    \indent When $R(\mathbf{A}) = n$, there is a unique solution. \\
    \indent When $R(\mathbf{A}) \neq R(\mathbf{A|b})$, there is no solution. \\
    \indent When $R(\mathbf{A}) = R(\mathbf{A|b}) < n$, there are infinitely many solutions. \\
    \\

    An \emph{eigenvector} is a vector whose direction remains unchanged after multiplication by a matrix. It is described by the \emph{characteristic equation}:
    \begin{equation}
        \mathbf{AX} = \lambda\mathbf{X}.
    \end{equation}
    The \emph{eigenvalues} $\lambda$ are values of $\lambda$ for which a non-trivial solution ($\mathbf{X} \neq 0$) exists:
    \begin{equation}
        \det(\mathbf{A} - \lambda\mathbf{I}) = 0.
    \end{equation}
    The \emph{Spectral Matrix} $\mathbf{S}$ contains the eigenvalues along the diagonal. \\
    The \emph{Modal Matrix} $\mathbf{M}$ contains the eigenvectors as columns.
    \begin{equation}
        \mathbf{A} = \mathbf{MSM}^{-1}
    \end{equation}

    Some properties of eigenvectors and eigenvalues are:
    \begin{align}
        & \eig \mathbf{A}\alpha = \alpha(\eig\mathbf{A}). \\
        & \eig \mathbf{A}^n = (\eig \mathbf{A})^n. \\
        & \eig \mathbf{A}^\intercal = \eig \mathbf{A}. \\
        & \eig (\mathbf{A} \pm k\mathbf{I}) = \(\eig\mathbf{A}) \pm k. \\
        & \eig \mathbf{A}^{-1} = \frac{1}{\eig \mathbf{A}}. \\
        & \eig(-\mathbf{A}) = -\eig\mathbf{A}.
    \end{align}
    \begin{align}
        & \sum \lambda_{\mathbf{A}} = \trace \mathbf{A}. \\
        & \prod \lambda_{\mathbf{A}} = \det(A).
    \end{align} \\




    \subsection{Arithmetic and Geometric Sequences and Series}

    An \emph{Arithmetic Sequence} has a common difference between elements, and is given by:
    \begin{equation}
        U_n = a + (n-1)d.
    \end{equation}
    Or by the recurrence relation:
    \begin{equation}
        x_{n+1} = x_n + d
    \end{equation}
    where $a$ is the first element and $d$ is the common difference. \\
    The \emph{Arithmetic Series} sums the Arithmetic Sequence and is given by
    \begin{equation}
        S_n = \sum_{k=0}^n U_k = \frac{1}{2}\left[2a + (n-1)d\right].
    \end{equation}
    The special case for $\mathbb{N}$ is:
    \begin{equation}
        S_n = \sum_{k=0}^n k = \frac{1}{2}(n+1).
    \end{equation} \\
    \\

    A \emph{Geometric Sequence} has a common ratio between elements, and is given by:
    \begin{equation}
        U_n = a \cdot r^{n-1}.
    \end{equation}
    Or by the recurrence relation:
    \begin{equation}
        x_{n+1} = rx_n.
    \end{equation}
    The \emph{Geometric Series} sums the Geometric Sequence and is given by:
    \begin{equation}
        S_n = \sum_{k=0}^n U_k = \frac{a(1-r^n)}{1-r}. \qquad r \neq 1
    \end{equation} \\
    The sum to infinity for a Geometric Series is given by:
    \begin{equation}
        S_\infty = \lim_{n\to\infty} \frac{a(1-r^n)}{1-r} = \frac{a}{1-r}. \qquad -1 < r < 1
    \end{equation}

    The sums of $k^2$ and $k^3$ are given by:
    \begin{eqnarray}
        \sum_{k=1}^n k^2 = \frac{1}{6} n(n+1)(2n+1).
        \\
        \sum_{k=1}^n k^3 = \left(\frac{n(n+1)}{2}\right)^2.
    \end{eqnarray} \\ 

    A series $S_n = \sum_{k=0}^n U_k$ diverges if $\Lim{n\to\infty} U_n \neq 0$. \\
    A sequence $U_n$ converges if $\Lim{n\to\infty} a_n = L$, \\
    \indent or diverges if it has no limit. \\
    \\

    The \emph{Radius of Convergence} $R$ is:
    \begin{equation}
        R = \lim_{n\to\infty} \left| \frac{a_n}{a_{n+1}} \right|.
    \end{equation} \\
\todo{todo: more divergence and convergence tests, absolute convergence}

    The \emph{Fibonacci} sequence is given by the recurrence relation:
    \begin{equation}
        F_{n+1} = F_n + F_{n-1}.
    \end{equation} \\
    \\

\clearpage
\addcontentsline{toc}{section}{Index}
\printindex

\end{document}

example argand.png

IleadsV.png

Q,I,t.png

VleadsI.png