
\part{Differentiable curves and surfaces}

\chapter{Differentiable curves}

\section{Differentiable curve}

We will define differentiable curves by means of \emph{position functions}; functions that represent one's position in a space.

\begin{definition}
A \emph{position function} is a vector valued function $\mathbf{r}: I \to \mathbb{X}^n $ that maps a real interval $I$ to some space $X^n$, essentially representing a path in a space. The argument of a position function will be referred to as \emph{time} since the parameter often represents time.
\end{definition}

As one could imagine, position functions are crucial in physics as physicists are often concerned with how forces modify the 'position' of objects.

As intuition may suggest, this function is 'nice' when it is continuous (the position doesn't magically 'teleport') and differentiable (we can calculate how much the direction of this position is changing with respect to time). Thus we arrive at the idea of a \emph{differentiable curve}.


\begin{definition}
	A \emph{differentiable curve} is a vector valued function $\boldsymbol{\gamma} : I \to \mathbb{R}^n $ that maps a real interval $I$ to some space $X^n$, essentially representing a path in a space.

	\[\boldsymbol{\gamma}'(t_0) = \lim_{t \to t_0} \frac{\boldsymbol{\gamma}(t) -\boldsymbol{\gamma}(t_0) }{t -t_0} \]
\end{definition}


\begin{definition}
A \emph{differentiable curve} is a position function $\mathbf{r} : I \to X$ that is totally differentiable. The image of this function is also called the differentiable curve. $X$ is some space, technically a differentiable manifold.
\[\mathrm{r} : [t_0,t_1] \to X \text{ is closed } \iff \mathbf{r}(t_0) = \mathbf{r}(t_1) \]
\end{definition}




A curve that is continuous but not differentiable is known as a \emph{topological curve}; this is a notion studied further in general topology.


\section{Closed differentiable curve}
\section{Line integral (scalar field)}


\begin{definition}
Let $\mathbf{r} : [t_0,t_1] \to \mathcal{C}$ be a differentiable curve. The \emph{arc length function of $\mathbf{r}$} is the function $s : [t_0,t_1] \to [0,\infty)$ that returns the length of the differentiable curve at a given time.
	\[s_{\mathbf{r}}(t) = \int^{t}_{t_0} \|\mathbf{r}'(t')\| dt'\]
\begin{definition}


A simple application of the FTC gives the following lemma.
\begin{lemma}
\[s'(t)=\| \mathbf{r}'(t)\|\]
\end{lemma}

\begin{definition}
A \emph{line integral (scalar field)} is an integral of a scalar field taken along a differentiable curve. One can think about it as curve's arc length the scaled by the scalar field's intensity along it.
\[ \int_{\mathcal{C}} f(\mathbf{x})ds = \int_{t_0}^{t_1} f(\mathbf{r}(t)) \| \mathbf{r}'(t) \|dt \]
\begin{itemize}
	\item $f : \mathbb{R}^n \to \mathbb{R}$ is a scalar field
	\item $\mathbf{r} : [t_0,t_1] \to \mathcal{C}$ is a differentiable curve parametrization
\end{itemize}
\end{definition}
Note how the previous lemma allows for the change of variables from $s$ to $t$ so that the line integral can be represented as an integral on an interval.

When representing a line integral along a closed curve, we will use the integral sign $\oint$ instead of $\int$



\begin{definition}
A \emph{line integral (vector field)} is an integral of a vector field taken along a differentiable curve. One can think about it as curve's arc length the scaled by the curve's dot product with the vector field's vectors along it. It is the 'work' of a vector field along that curve.
\[ \int_{\mathcal{C}} \mathbf{f}(\mathbf{x}) \cdot d\mathbf{r} = \int_{\mathbf{r}(t_0)}^{\mathbf{r}(t_1)} \mathbf{f}(\mathbf{r}(t)) \cdot \mathbf{r}'(t)dt \]
\begin{itemize}
	\item $\mathbf{f} : \mathbb{R}^n \to \mathbb{R}$ is a vector field
	\item $\mathbf{r} : [t_0,t_1] \to \mathcal{C}$ is a differentiable curve parametrization
\end{itemize}
\end{definition}


\chapter{Differentiable surfaces}
\section{Differentiable surface}
\section{Various parametrization}
\section{Surface integral}
\section{Flux integral}


