Section Summary: 16.8

Triple Integrals in Cylindrical and Spherical Coordinates

1. Definitions

2. Theorems

   If

   

   where

   

   then

   

   If

   

   then

   

3. Properties/Tricks/Hints/Etc.

   Note that the element of volume dV for cylindrical coordinates is a simple generalization of that for polar coordinates:

   

   For spherical coordinates,

   

   These ``inflation/deflation'' factors will be derived in section 16.9.

4. Summary

   Changing coordinate systems is not done frivolously, but generally because of some symmetry properties of a problem that make the problem easier to represent (or to solve) in that system.

   Coordinate systems are ``known'' (in some sense) by the equations obtained by setting each coordinate variable equal to a constant. In rectangular coordinates, mutually orthogonal planes result. In the case of cylindrical coordinates, the three surfaces are two mutually orthogonal planes and a cylinder. Where they intersect in a point, the three surfaces are (locally) mutually orthogonal.

   For spherical coordinates, the three surfaces are a plane, a spherical shell, and a cone: again, at their intersection in a point, they are locally mutually orthogonal.