example of integration with respect to surface area on a sphere viewed as a graph
In this example, we consider the sphere as the graph of and use equation (1) of the http://planetmath.org/node/6660main entry to express integrals over the sphere with respect to surface area as double integrals with respect to and .
Differentiating the function and simplifying, we obtain
Hence, we have
This formula is sometimes written as
In using this form, it is understood that is to be regarded as a function of and rather than as an independent variable.
1 Note on multi-valuedness
To use this formula correctly, one must pay attention to the fact that the square root is multiply valued – to every pair of values with , there correspond two values of . If one chooses only one branch of the square root (say the negative branch), one will only take into account half of the surface area of the sphere (in the case of the negative branch, this would be “southern hemisphere” which lies below the plane). Therefore, unless one is only interested in carrying out an integral over a single hemisphere, one needs to account for both points on the sphere that correspond to a point in the plane. A more careful way of rewriting this formula which takes this into account would be
Here, denotes the value of the function at the point on the sphere and, likewise, denotes the value of at the point . However, the formula is usually not written in this way, and it is left up to the reader to remember that both hemispheres must be accounted for.
2 Polar coordinates
At times, it is useful to describe the sphere as a graph and use polar coordinates in the plane. To adapt our formula to this situation, we need to make the change of variables
The Jacobian for this transform is
and hence we have
To return to the main entry, please http://planetmath.org/node/6660click here. To go back to example 4, please http://planetmath.org/node/6667click here.
|Title||example of integration with respect to surface area on a sphere viewed as a graph|
|Date of creation||2013-03-22 14:58:08|
|Last modified on||2013-03-22 14:58:08|
|Last modified by||rspuzio (6075)|