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8.3 When is a Vector Field the Curl of another?

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 = 0.
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.