spherical polar coordinates

Changing x,y,z coordinates into spherical polar coordinates

If you need to do calculations involving a shape which is spherically symmetric then the working will be much easier if you use spherical polar coordinates to describe each point.
The drawing below shows these coordinates for the point P. The origin, O, is chosen so that it is the centre of spherical symmetry.
how the spherical polar coordinates are measured

how the spherical polar coordinates are measured

Your calculation is quite likely to involve some function given in terms of the x, y and z coordinates of the point P. In this case you will need to know how to convert the x, y and z into spherical polar terms.
The drawing below shows you how to do this.
changing x,y,z coordinates into spherical polar coordinates

changing x,y,z coordinates into spherical polar coordinates

Spherical polar coordinates and triple integrals

Suppose you need to find the volume of a body. You could do this by splitting the body up into lots of tiny rectangular boxes and then adding the volumes of all the little boxes together. Letting the sides of each box become infinitesimally small and the number of boxes become infinitely large will give you the triple integral for the volume of

over the body which you are considering.
If this body is spherically symmetric, the working out is much simpler if you replace the little rectangular boxes with the little curved boxes which you get if you let the point P move so that each of its 3 polar coordinates increases separately by an infinitesimally tiny bit.
The picture below shows this, considering each coordinate in turn and somewhat enlarged so that you can actually see what's happening .
the effect of a tiny increment in each of the spherical polar coordinates

the effect of a tiny increment in each of the spherical polar coordinates

Now, if P moves so that all 3 of its polar coordinates increase, you'll get a little slightly curved box.

the tiny box made by slightly increasing all 3 coordinates

dV is its infinitesimally small volume. We can find it by multiplying the sides together because, if the box is sufficiently tiny, it can be considered to be a little rectangular box. All the angles at the corners are right angles. So to find the total volume, instead of summing over all the infinitesimally tiny exactly rectangular boxes with volume dV = dx dy dz, you can say

Now, suppose you are told some function like density, for example, which is defined in terms of the x,y,z coordinates of each point P of some spherically symmetric body. We'll call it f(x,y,z). In order to find the total mass of the body, you'll first have to convert all the x, y and z terms of this function into spherical polar terms.
Suppose that doing this gives you F(r,q,f). This should read as F(r,theta,phi) as a check that it is O.K on all browsers.
Then the total mass M is given by

integrated over the body you are considering.

return to the vectors homepage