Integral Theorems and Distributions

In classical mechanics and electrod)mamics the integral theorems of Gauss and Stokes may often be employed beneficially. Given a sufficiently smooth (i.e., differentiable) and well-behaved vector field A, Gauss theorem may be expressed in its most elementary form as 

Here a again denotes the outer normal unit vector perpendicular to the surface S. Eq. (A.20) is sometimes also referred to as the curl theorem. 

In discussions of the Coulomb potential the following relations are often useful. The gradients of a distance between two positions, r and r, read 

The indices in Eq. (A.21) label the variable with respect to which the derivative is taken. We note that the proof of Eq. (A.24) is only trivial for r — r 0 for which the right-hand side becomes zero. In this case, where r /, we may simply differentiate the inverse distance and obtain this result right away. 

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