There is one essential theorem of vector calculus that is essential to the development of multipoles - computing the dipole moment. Jackson blithely integrates by parts (for a charge/current density with compact support) thusly:
Then, using the continuity equation and the fact that and
are presumed harmonic with time dependenct
, we substitute
However, this leaves a nasty question: Just how does this integration by parts work? Where does the first equation come from? After all, we can't rely on always being able to look up a result like this, we have to be able to derive it and hence learn a method we can use when we have to do the same thing for a different functional form.
We might guess that deriving it will use the divergence theorem (or Green's theorem(s), if you like), but any naive attempt to make it do so will lead to pain and suffering. Let's see how it goes in this particularly nasty (and yet quite simple) case.
Recall that the idea behind integration by parts is to form the
derivative of a product, distribute the derivative, integrate, and
The exact same idea holds for vector calculus, except that the idea is
to use the divergence theorem to form a surface integral
instead of a boundary term. Recall that there are many forms of the
divergence theorem, but they all map
following integral form:
To prove Jackson's expression we might therefore try to find a suitable
product whose divergence contains
as one term. This isn't too
easy, however. The problem is finding the right tensor form. Let us
look at the following divergence:
To assemble the right answer, we have to sum over the three separate statements:
We rearrange this and get:
This illustrates one of the most difficult examples of using integration by parts in vector calculus. In general, seek out a tensor form that can be expressed as a pure vector derivative and that evaluates to two terms, one of which is the term you wish to integrate (but can't) and the other the term you want could integrate if you could only proceed as above. Apply the generalized divergence theorem, throw out the boundary term (or not - if one keeps it one derives e.g. Green's Theorem(s), which are nothing more than integration by parts in this manner) and rearrange, and you're off to the races.
Note well that the tensor forms may not be trivial! Sometimes you do have to work a bit to find just the right combination to do the job.