In a previous section we considered the question: when is a vector field the
gradient of a potential: the answer was (in a simply connected region)
when the vector field is curl free.
We now ask: when can a vector field be written as the curl of another? The answer is equally simple, in a simply connected region:
We can write v =A
in R, R simply connected,if and only if v
is divergence free in R:v
= 0 in R. When this occurs, we call A a vector potential for v in R.
Again, this condition is obviously necessary.
If v = A holds thenv = A.
The vector operator ((consists
of six terms, the three cross partials and their negatives. It is identically
zero and therefore we have v
This claim has an important implication. It means we can write any suitably well behaved vector field v as the sum of the gradient of a potential f and the curl of a vector potential A. One can produce its divergence with curl 0, and the other can supply its curl with divergence 0: any such vector field v can be written as
v = f +A
We will prove this claim in the important special case in which R is the entire
space, and do so constructively: we actually produce useful formulae for f and
A, given that it approach a constant at infinity.
Since only the gradient of f is relevant to v, v only determines f up to an additive constant. Similarly, since only the curl of A is relevant to v, one can add the gradient of any function g to A without changing v. Modifying A in such a way is called 'making a gauge transformation'. Specifying the divergence of A is called 'choosing a gauge'. Obviously v is not affected by gauge transformations.
The proof and formulae will be given in a later section. It is based upon three ideas:
1. We know the potential produced by a point charge in electrostatics. It is given by Coulomb's law.
2. The potential produced by a given distribution of charge can be obtained by summing or integrating Coulomb's law over the distribution.
3. The double cross product identity applied to the vector operator gives us a way to apply these facts to determine f and A.