Spherical Coordinates

#Math
Wolfram Article

$\displaystyle (\rho,\theta,\phi)=\left( \sqrt{ x^{2}+y^{2}+z^{2} },\tan ^{-1}\left( \frac{y}{x}\right),\cos ^{-1}\left( \frac{z}{\rho} \right) \right)$

• Conversion from rectangular coordinates to spherical coordinates
• $\displaystyle \theta$ is the polar angle
• $\displaystyle \phi$ is the azimuthal angle
• The $\displaystyle \theta$ and $\displaystyle \phi$ symbols are often swapped in the context of physics

$\displaystyle (x,y,z)=(\rho \sin \phi \cos \theta,\rho \sin \phi \sin \theta,\rho \cos \phi)$

• Conversion from spherical to rectangular coordinates

$\displaystyle J_{\text{spherical}}=\rho^{2}\sin\phi$

• Jacobian for conversion from rectangular to spherical coordinates