07 - Coordinate Systems

Plane Polar Coordinates

For $P=(x,y)\in {\mathbb{R}}^2$ and $(x,y)\ne (0,0)$ then

Let $r$ be the distance from $(0,0)$ and
Let $\theta$ be the anti-clockwise angle that $\theta$ that $\overrightarrow{OP}$ makes with the $x-$axis

Hence

$$x=r\cos \theta ,y=r\sin \theta$$

Where $r\in [0,\infty )$ ad $\theta \in [0,2\pi )$

Jacobian

Jacobian Matrix

$$\left(\begin{array}{cc}\frac{\partial x}{\partial r}& \frac{\partial x}{\partial \theta }\\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial \theta }\end{array}\right)=\left(\begin{array}{cc}\cos \theta & -r\sin \theta \\ \sin \theta & r\cos \theta \end{array}\right)$$

Jacobian

$$\frac{\partial (x,y)}{\partial (r,\theta )}=r$$

Parabolic Coordinates

For coordinates $u,v$ given in terms of $x,y$

$$x=\frac{1}{2}(u^2-v^2)y=uv$$

Jacobian

$$\frac{\partial (x,y)}{\partial (u,v)}=\left|\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial x}{\partial v}\\ & \\ \frac{\partial y}{\partial u}& \frac{\partial y}{\partial v}\end{array}\right|=\left|\begin{array}{cc}u& -v\\ v& u\end{array}\right|=u^2+v^2$$

Cylindrical Polar Coordinates

Extend plane polar coordinates to three dimensions by $z$-coordinates
So that

$$x=r\cos \theta ,y=r\sin \theta ,z=z$$

Inverse Transformation

$$r=\sqrt{x^2+y^2},\tan \theta =\frac{y}{x},z=z$$

Jacobian

Jacobian Matrix

$$\left(\begin{array}{ccc}\frac{\partial x}{\partial r}& \frac{\partial x}{\partial \theta }& \frac{\partial x}{\partial z}\\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial \theta }& \frac{\partial y}{\partial z}\\ \frac{\partial z}{\partial r}& \frac{\partial z}{\partial \theta }& \frac{\partial z}{\partial z}\end{array}\right)=\left(\begin{array}{ccc}\cos \theta & -r\sin \theta & 0\\ \sin \theta & r\cos \theta & 0\\ 0& 0& 1\end{array}\right)$$

Hence the Jacobian is

$$\frac{\partial (x,y,z)}{\partial (r,\theta ,z)}=r{\cos }^2\theta +r{\sin }^2\theta =r$$

Spherical Polar Coordinates

Let $(x,y,z)$ be the Cartesian Coordinates for a general point $P\in {\mathbb{R}}^3$

For $P\ne (0,0,0)$
Let $r$ be the distance between $P$ and $O$
Let $\theta$ be the angle from the z-axis to the position vector $\overrightarrow{OP}$
Let $\varphi$ be the angle when you convert $(x,y)$ to coordinates $(\rho \cos \varphi ,\psi \sin \varphi )$

Hence in terms of the spherical coordinates $(r,\theta ,\varphi )$ we have

$$x=r\sin \theta \cos \varphi ,y=r\sin \theta \sin \varphi ,z=r\cos \theta$$

where $r\ge 0$, $0\le \theta \le \pi$ and $0\le \varphi <2\pi$

Inverse Transformation

$$r=\sqrt{x^2+y^2+z^2},\tan \theta =\frac{\sqrt{x^2+y^2}}{z},\tan \varphi =\frac{y}{x}$$

Jacobian

Jacobian Matrix

$$\left(\begin{array}{ccc}\frac{\partial x}{\partial r}& \frac{\partial x}{\partial \theta }& \frac{\partial x}{\partial \varphi }\\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial \theta }& \frac{\partial y}{\partial \varphi }\\ \frac{\partial z}{\partial r}& \frac{\partial z}{\partial \theta }& \frac{\partial z}{\partial \varphi }\end{array}\right)=\left(\begin{array}{ccc}\sin \theta \cos \varphi & r\cos \theta \cos \varphi & -r\sin \theta \sin \varphi \\ \sin \theta \sin \varphi & r\cos \theta \sin \varphi & r\sin \theta \cos \varphi \\ \cos \theta & -r\sin \theta & 0\end{array}\right)$$

Jacobian

$$\frac{\partial (x,y,z)}{\partial (r,\theta ,\varphi )}=r^2\sin \theta$$