Today, we try to evaluate the following surface integral in . I found this result in a paper published in Math. Z. 198, 277-289 (1988). This topic can be considered as a continued part to the following topic.
Proposition. Let be a continuous function. Then we have
here we denote .
Proof. Remark that the integral only depends on , so that we may assume and introduce the following spherical coordinate
This system of coordinates can be seen via the picture below
Let us remember once a surface can be parametrized by two variables , we then have
Thus by the above coordinate
which helps us to write down
The last integral can be estimated further as follows
By a variable change
we arrive at
Therefore if we choose to be we obtain
Corollary. The following identity
It is well-known that
Thus we have the following formula for the average of
over a sphere
This formula can also be found on page 249 of a book due to Lieb and Loss (AMS 2001).