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\section*{Introduction}

We will learn to differentiate and integrate
functions (or maps) of the form
\[ f : \ub{\RR^m}_{\text{domain}} \to
\ub{\RR^n}_{\text{codomain}} \]
An element of $\RR^m$ or $\RR^n$ is a vector so
this subject is called \emph{vector calculus}.

\subsubsection*{Examples of Maps}

\begin{enumerate}[(1)]
    \item A function $f : \RR \to \RR^n$ defines a
        \emph{curve} in $\RR^n$. In physics, we
        might think of $\RR$ as time and $\RR^n$
        as physical space and write this as
        \[ f : t \mapsto \bf{x}(t) \]
        with $\bf{x} \in \RR^n$. (Obviously we
        should take $n = 3$). Generalising, a map
        \[ f : \RR^2 \to \RR^n \]
        defines a surface in $\RR^n$, and so on.
    \item In other applications, the domain
        $\RR^m$ might be viewed as physical space.
        For example, in physics a \emph{scalar
        field} is a map
        \[ f : \RR^3 \to \RR \]
        for example temperature $T(\bf{x})$ is a
        scalar field, as is the Higgs field. \\
        A \emph{vector field} is a map
        \[ f : \ub{\RR^3}_{\text{physical space}}
        \to \ub{\RR^3}_{\text{somethinge more
        abstract}} \]
        for example the electric field
        $\bf{E}(\bf{x})$ and magnetic field
        $\bf{B}(\bf{x})$ are vector fields.
\end{enumerate}

\newpage
\section{Curves}

We consider maps of the form
\[ f : \RR \to \RR^n \]
Assign a coordinate $t$ to $\RR$ and use Cartesian
coordinates on $\RR^n$.
\[ \bf{x} = (x^1, \dots, x^n) = x^i \bf{e}_i \]
where $\bf{e}_i$ is an orthonormal basis such that
$\bf{e}_i \cdot \bf{e}_j = \delta_{ij}$. Note that
summation convention is used here. (For $\RR^3$ we
also use notation $\{\bf{e}_i\} = \{\hat{\bf{x}},
\hat{\bf{y}}, \hat{\bf{z}}\}$.)

\bigskip

\noindent
The image of of the function $f$ is a
\emph{parametrised curve} $\bf{x}(t)$, with $t$
the parameter.

\subsubsection*{Examples}

\begin{enumerate}[(1)]
    \item Consider the map $\RR \to \RR^3$ given
        by
        \[ \bf{x}(t) = (at, bt^2, 0) \]
        The curve $C$ is the parabola $a^2y = bx^2$
        in the plane $z = 0$.
        \begin{center}
        \begin{tsqx}
            (-1,0)->>(1,0)
            (0,-1)->>(0,1)
            ! real f(real x) {return x**2;}
            ! draw(graph(f, -1, 1, operator, ..), green);
            label $x$ @ (1.1,0)
            label $y$ @ (0,1.1)
            (0.5,0.2)->>(-0.5,-0.2)
            label $z$ @ 1.1*(-0.5,-0.2)
        \end{tsqx}
        \end{center}
        \begin{note*}
            When plotting the curve, we lose
            information about the parameter $t$.
        \end{note*}
    \item Consider $\bf{x}(t) = (\cos t, \sin t,
        t)$
        \begin{center}
            \includegraphics[width=0.6\linewidth]
            {images/fdfa3aea7aa611ec.png}
        \end{center}
        The curve $C$ is a helix.
\end{enumerate}
The choice of parametrisation is not
unique, for example consider
\[ \bf{x}(t) = (\cos \lambda t, \sin
\lambda t, \lambda t) .\]
This gives the same helix for all $\lambda
\in \RR \setminus \{0\}$. \\
Sometimes the choice of parametrisation
matters, for example if $t$ is time and
$\bf{x}(t)$ is position, then the velocity
is proportional to $\lambda$. But we will
see that some questions are independent of
the choice of parametrisation.

\subsection{Differentiating the Curve}

A vector function $\bf{x}(t)$ is differentiable of
$t$ if, as $\delta t \to 0$, we have
\[ \bf{x}(t + \delta t) - \bf{x}(t) =
\dot{\bf{x}(t)} \delta(t) + O(\delta t^2) .\]
If $\dot{\bf{x}}(t)$ exists everywhere, the curve
is said to be \emph{smooth}.
\begin{note*}
    ``Big $O$'' notation $O(\delta t^2)$ means
    terms proportional to $\delta t^2$ or smaller.
\end{note*}
In physics, dot is usually used for time
derivatives, for example $\dot{\bf{x}}(t)$ and
prime for spatial derivatives, for example
$f'(x)$. \\
In maths, these are used interchangeably.

\bigskip

\noindent
Some notation: we write
\[ \delta \bf{x}(t) = \bf{x}(t + \delta t) -
\bf{x}(t) \]
The derivative is then
\[ \bf{x} \equiv \dfrac{\bf{x}}{t} := \lim_{\delta
t \to 0} \frac{\delta \bf{x}}{\delta t} .\]