Spherical Coordinates

Spherical coordinates are a 3D coordinate system that can be thought of as applying polar coordinates twice. It is especially useful when the problem has complete rotational symmetry around all axis.

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Definition §

The spherical coordinate system is notated as:

$$(r,\theta ,\varphi )$$

• $r$ : The radius, must be a non-zero number. $r\ge 0$
• $\theta$ : The polar angle, starting from the positive $z$-axis and going towards the negative $z$-axis. Think of moving from the ‘North pole’ to the ‘South pole’. $\theta \in [0,\pi )$
• $\varphi$: The azimuthal angle, starting from the $x$-axis, going anti-clockwise (towards y-axis). $\varphi \in [0,2\pi )$

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Conversions §

With Cartesian Coordinates §

Every point in 3D Cartesian coordinates $(x,y,z)$ can be converted to spherical coordinates via the equation:

$$x=r\sin (\theta )\cos (\varphi )y=r\sin (\theta )\sin (\varphi )z=r\cos (\theta )$$

And vice versa:#todo

$$r=\Vert (x,y,z)\Vert =\sqrt{x^2+y^2+z^2}\theta =\arccos (\frac{z}{r})=\arccos \left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right)\varphi =\{\begin{array}{ll}\arctan \left(\frac{y}{x}\right),& x>0,y>0(\text{Q1})\\ \pi +\arctan \left(\frac{y}{x}\right),& x<0,y>0(\text{Q2})\\ \pi -\arctan \left(\frac{y}{x}\right),& x>0,y<0(\text{Q3})\\ 2\pi -\arctan \left(\frac{y}{x}\right),& x<0,y<0(\text{Q4})\end{array}$$

Key Features §

• If $r=c$ (radius is constant), then we have an sphere centred at the origin with radius $r$
• If $\theta =c$ (polar angle is constant), then we have an open cone parallel to the $z$-axis, with its vertex at the origin.
• If $\varphi =c$ (azimuthal angle is constant), then we have a half-plane parallel to the $XZ$ plane after a rotation $\varphi$. (Same as in the cylindrical coordinates case!)

Jacobian §

Cartesian Jacobian §

The Jacobian is given by:

$$\begin{array}{rl}\det (\mathbf{J})& =\det \left[\begin{array}{ccc}\frac{\partial x}{\partial r}& \frac{\partial x}{\partial \theta }& \frac{\partial x}{\partial \varphi }\\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial \theta }& \frac{\partial y}{\partial \varphi }\\ \frac{\partial z}{\partial r}& \frac{\partial z}{\partial \theta }& \frac{\partial z}{\partial \varphi }\end{array}\right]=\det \left[\begin{array}{ccc}\sin (\theta )\cos (\varphi )& r\cos (\theta )\cos (\varphi )& -r\sin (\theta )\sin (\varphi )\\ \sin (\theta )\sin (\varphi )& r\cos (\theta )\sin (\varphi )& r\sin (\theta )\cos (\varphi )\\ \cos (\theta )& -r\sin (\theta )& 0\end{array}\right]\\ & \\ & =\cos (\theta )[r^2\cos (\theta )\sin (\theta ){\cos }^2(\varphi )+r^2\sin (\theta )\cos (\theta ){\sin }^2(\varphi )]\\ & +r\sin (\theta )[r{\sin }^2(\theta ){\cos }^2(\varphi )+r{\sin }^2(\theta ){\sin }^2(\varphi )]+0\\ & =r^2{\cos }^2(\theta )\sin (\theta )+r^2{\sin }^3(\theta )\\ & =r^2\sin (\theta )\end{array}\text{\hspace{1em}(Expanding along 3rd row)}$$

Hence:

$${\iiint }_{D}f(x,y,z)\text{d}x\text{d}y\text{d}z={\iiint }_{D^{\ast }}f(r\sin (\theta )\cos (\varphi ),r\sin (\theta )\sin (\varphi ),r\cos (\theta ))\cdot r^2\sin (\theta )\text{d}r\text{d}\theta \text{d}\varphi$$