\chapter{Coordinate frames}

Readers will be familiar with coordinate systems.

Coordinate frame is a Basis that formed based on some arbitrary 'origin' coordinate. It encapsulates the idea 
For instance, at some coordinate $p$, one may want a frame representing the directions of the increase of each coordinate (cartesian, cylindrical, spherical frames), or perhaps one wants a frame with directions along a curve, orthogonal to the bend of a curve, and another direction orthogonal to both of those (Frenet-Serret frame).

Considering different coordinate frames can immensely reduce and facilitate the required computations for many types of integrals one encounters in vector analysis by abusing the symmetry of various problems, and is also vital in the study of differential geometry.


We first study the former; frames of coordinate systems.

Note that although many frames we use are orthonormal basis', but this is not a requirement; we'll see examples of these much later on.

coordinate frame
\begin{definition}
Let $x$ be a coordinate system and $\mathbf{r}$ map the coordinate system to $S$, then the \emph{orthonormal coordinate frame with respect to $p$} is the orthonormal basis of vectors defined by the set of $\mathbf{e}_{x}(p) = \frac{ \mathbf{r}_{x}(p) }{ \| \mathbf{r}_{x}(p) \| }$ for each coordinate.
\end{definition}


scale factor

divide by scale factor to make normalized coordinate frame



\section{Cartesian $\mathbb{R}^n$ frame}

Cartesian basis is the standard basis

Transform from cartesian basis to cartesian basis is done by identity linear transform







Cartesian frame

\[(x,y,z)\]

\[\mathbf{r}(x,y,z) = \begin{bmatrix} x \\ y \\ z \end{bmatrix}\]

	\[ \mathcal{F} =  \{  \mathbf{e}_{x} , \mathbf{e}_{y} ,  \mathbf{e}_{z} \}   = \{ \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} ,  \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix} ,  \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \} = \{ \mathbf{i} \mathbf{j}, \mathbf{k}\} \]


The cartesian frame has the interesting property in that the generated basis is the same relative to any coordinate point. As it turns out, other frames do not have this property. 









\section{Generalizing to a curvilinear orthonormal frame}

The cartesian coordinate system is the standard, we can define new curvilinear coordinate systems by introducing invertible differentiable functions that translate the new coordinates tocartesian coordinates.


From any coordinate system, we define its coordinate frame by the following.

This normalization of a linear transform is essentially saying that the basis for the new coordinate frame has vectors representing the direction of infinitesimal increase for each coordinate at some given point. This means that the coordinate frame can have a whole new basis for different points. We normalize it for the sake of making the vectors have length 1.

As it turns out, the basis vectors are always orthogonal if our cartesian to new coordinate functions are differentiable, so we have an orthonormal basis!



\section{Polar $\mathbb{R}^2$ basis}
\[x(r,\theta) = r \cos (\theta)\]
\[y(r,\theta) = r \sin(\theta)\]

According to the definition of a coordinate frame when these functions are differentiable and invertible (which they are on $r\in (0,\infty) ,\theta \in [0,2\pi)$), we have the following by applying the linear transform an then normalizing.

\[ \mathbf{r} = \cos (\theta) \mathbf{x} + \sin (\theta) \mathbf{y} \]
\[ \mathbf{\theta} = -\sin (\theta) \mathbf{x} + \cos(\theta) \mathbf{y} \]


\section{Cylindrical $\mathbb{R}^3$ basis}



Cylindrical frame
\[(r,\theta,z)\]
\[\mathbf{r}(r,\theta,z) = \begin{bmatrix} r \cos (\theta)  \\ r \sin (\theta) \\ z \end{bmatrix}\]

\[ \mathcal{F} =  \{  \mathbf{e}_{r} , \mathbf{e}_{\theta} ,  \mathbf{e}_{z} \}   = \{ \begin{bmatrix} \cos (\theta) \\ \sin (\theta) \\ 0 \end{bmatrix} ,  \begin{bmatrix} -\sin (\theta) \\ \cos (\theta) \\ 0 \end{bmatrix} ,  \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \} \]




\section{Spherical $\mathbb{R}^3$ basis}


Spherical frame
\[(\rho,\theta,\phi)\]
\[ \mathbf{r}(\rho,\theta,\phi) = \begin{bmatrix} \rho \sin (\phi) \cos (\theta)  \\ \rho \sin (\phi) \sin (\theta) \\ \rho \cos (\phi) \end{bmatrix}\]
	\[ \mathcal{F} =  \{  \mathbf{e}_{\rho} , \mathbf{e}_{\theta} ,  \mathbf{e}_{\phi} \}   = \{ \begin{bmatrix} \sin (\phi) \cos (\theta) \\ \sin (\phi) \sin (\theta) \\ \cos (\phi) \end{bmatrix} ,  \begin{bmatrix} \cos (\phi) \cos (\theta) \\ \cos (\phi)  \sin (\theta) \\ -\sin (\phi) \end{bmatrix} ,  \begin{bmatrix} \cos (\phi) \\ -\sin (\phi) \\ 0 \end{bmatrix} \} \]



\section{Applications for line integrals}
\section{Applications for surface and flux integrals}





































\chapter{Frenet-Serret frame}


Say we want to make an orthonormal frame not based on coordinates of points, but based on position along a differentiable curve with the direction of its tangent, its direction of acceleration and then direction orthomormal to both of those.

For some differentiable curve $\mathbf{r}$, $\mathbf{r}'$ represents it's 'tangent'; the direction that the curve is currently 'facing'. We'd love a normalized version of this to be in this 'differentiable curve frame' of ours. 

We'd also like its 'acceleration' $\mathbf{r}''$; this tells us in which direction the curve is 'curving' towards. However is taking the derivative of a vector function sufficient to give us a new orthogonal vector functions?

\begin{proposition}
\[\mathbf{f} \perp \mathbf{f}'\]
\end{proposition}

So far so good, we normalize this 'acceleration' vector and add it to our frame. The last member of the frame can be done by taking the cross product of the 2 vectors; since they are both normalized, the result of this is also a normal vector.

The frame we've just described is called the \emph{Frenet-Serret frame}.

\section{Frenet-Serret frame}

\begin{definition}
Let $\mathbf{r}$ be a differentiable curve, the \emph{Frenet-Serret frame of $\mathbf{r}$} is an orthonormal basis spanning \(\mathbb{R}^3\) relative to a a point on the differentiable curve \(\mathbf{r}(s)\) that is parametrized by arclength \(s\).

	\[ \mathbf{T}(s) := \mathbf{r}'(s)\]
	\[ \mathbf{N}(s) := \frac{\mathbf{T}'(s)}{\| \mathbf{T}'(s) \|}\]
	\[ \mathbf{B}(s) := \mathbf{T}(s) \times \mathbf{N}(s)\]
	$\mathbf{T}$ (tangent) direction tangent to the differentiable curve
	$\mathbf{N}$ (normal) direction the differentiable curve is turning into against its tangent
	$\mathbf{B}$ (binormal) direction orthogonal to $\mathbf{T}, \mathbf{N}$

\end{definition}

By the chain rule, one can express the Frenet-Serret frame with respect to time rather than arclength.

\begin{proposition}
	\[ \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\mathbf{r}'(t)}\]
	\[ \mathbf{N}(t) = \frac{\mathbf{T}'(t)}{\| \mathbf{T}'(t) \|}\]
	\[ \mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)\]
\end{proposition}

For developing mathematical theory, it is more natural to deal with arclength parametrizations, however calculations may be easier with respect to time instead. We'll therefore continue using arclength parametrizations for theory and use time parametrizations in concrete examples.







\begin{definition}
The \emph{curvature of a differentiable curve} is the measure of directional change on the tangential plane.
\[ \kappa (s) = \| \mathbf{T}'(s) \| \]
\[ \kappa (t) = \frac{\| \mathbf{T}'(t) \|}{\| \mathbf{r}'(t)\|} \]
\end{definition}


\begin{definition}
The \emph{torsion of a differentiable curve} is the measure of directional change out of the tangential plane.
\[ \tau(s) = \| \mathbf{B}'(s) \| \]
\[ \tau (t) = \frac{\| \mathbf{B}'(t) \|}{\| \mathbf{r}'(t)\|} \]
\end{definition}

\begin{theorem}[Frenet-Serret formulae]
\[ \mathbf{T}'(s) = \kappa(s) \mathbf{N}(s) \]
\[ \mathbf{N}'(s) = -\kappa(s) \mathbf{T}(s) +\tau(s) \mathbf{B}(s) \]
\[ \mathbf{B}'(s) = -\tau(s) \mathbf{N}(s) \]


\[ \begin{bmatrix} \mathbf{T}'(s) \\  \mathbf{N}'(s) \\ \mathbf{B}'(s) \end{bmatrix} = \begin{bmatrix} 0 & \kappa(s) & 0 \\ -\kappa(s) & 0 \tau (s) \\ 0 & -\tau(s) & 0 \end{bmatrix} \begin{bmatrix}  \mathbf{T}(s)  \\ \mathbf{N}(s) \\ \mathbf{B}(s) \end{bmatrix}\]
\[  = -\kappa(s) \mathbf{T}(s) +\tau(s) \mathbf{B}(s) \]
\[ \mathbf{B}'(s) = -\tau(s) \mathbf{N}(s) \]

\end{theorem}



\section{Curvature}

\[ \kappa(t) = \frac{\| \mathbf{r}'(t) \times \mathbf{r}''(t) \|}{\| \mathbf{r}'(t)\|^{3}} \]
\section{Torsion}
\[ \tau(t) = \frac{ ( \mathbf{r}'(t) \times  \mathbf{r}''(t) )\cdot \mathbf{r}'''(t)  }{\| \mathbf{r}'(t) \times \mathbf{r}''(t)\|^{2}}\]


\begin{itemize}
\item Osculating plane $\mathrm{span}\{ \mathbf{T} , \mathbf{N} \}$
\item Rectifying plane $\mathrm{span}\{ \mathbf{T} , \mathbf{B} \}$
\item Normal plane $\mathrm{span}\{ \mathbf{N} , \mathbf{B} \} $
\end{itemize}