Question

Write the equations for converting from spherical to rectangular coordinates. In what situation would you use spherical coordinates?

Short answer

The required equations are

\(\begin{aligned}{} & x=\rho \cos \theta \sin \phi \\{} & y=\rho \sin \theta \sin \phi \\{} & z=\rho \cos \phi \\\end{aligned}\)

Step-by-step solution

Convert the coordinates

To convert from spherical \(\left( \rho ,\theta ,\varphi \right)\) to rectangular coordinates \(\left( x,y,z \right)\), we use the equations

\(\begin{aligned}{} & x=\rho \cos \theta \sin \phi \\{} & y=\rho \sin \theta \sin \phi \\{} & z=\rho \cos \phi \\\end{aligned}\)

The spherical coordinate system is especially useful in problems where there is symmetry about a point, and the origin is placed at this point.

Diagram of spherical coordinates

The spherical coordinate system is especially useful in problems where there is symmetry about a point, and the origin is placed at this point.
For example, the sphere with the centre at the origin and radius \(c\) has the simple equation \(\rho =c\).