Spherical coordinates

Spherical coordinates are a way to describe the location of a point in three-dimensional space based on <math>r</math>, the distance from the origin (where <math>x=y=z=0</math>); <math>\theta</math>, the angle between the point and the x-z plane (in the positive x direction); and <math>\phi</math>, the angle between the point and the z axis (in the positive z direction). Thus, each point is described by: <math>(r, \theta, \phi)</math>.

In a sense, spherical coordinates are coordinates on a sphere just like polar coordinates are coordinates on a circle.

The equations converting the parameters are as follows:

<math>r^2 = x^2 + y^2 + z^2</math>

<math>x = r \sin{\theta} \cos{\phi}</math>

<math>y = r \cos{\theta} \sin{\phi}

<math>z = r \cos{\phi}</math>

The Jacobian in spherical polar coordinates is <math>r^2 \sin{\theta}</math> so that <math>\text{d}x \, \text{d}y \, \text{d}z = r^2 \sin{\theta} \, \text{d}r \, \text{d} \theta \, \text{d} \phi</math>.