Application to differential geometry

Let

.

Differentiate with respect to and then integrate the result with respect to .

The above expression was posted to a message board of my homepage from some person. I deleted the message board in my error. I suppose that the date of submission was around 2001. I do no know his/her name.

Now, the left side of the expression can be drawn as a difference of volume of rectangular parallelepipeds as shown below.