Vector potentials

How many vector fields admit a certain vector potential?

Original post by JrW13

How many vector fields admit a certain vector potential?

Do you mean: given

A

, for how many fields

B

is it true that

B=\nabla \times A

?

OP

Yes

Original post by JrW13

How many vector fields admit a certain vector potential?

Recall, as we saw in this thread, that

\displaystyle \nabla \times A = \nabla \times (A + \nabla \phi)

for some scalar function

\phi

. Now recall that a vector field is irrotational if and only if it admits a scalar potential. Put these two facts together and you get the answer.

Original post by Gregorius

Recall, as we saw in this thread, that

\displaystyle \nabla \times A = \nabla \times (A + \nabla \phi)

for some scalar function

\phi

. Now recall that a vector field is irrotational if and only if it admits a scalar potential. Put these two facts together and you get the answer.

You seem to have answered the inverse question, AFAICS: how many vector potentials does a certain vector field admit?

Original post by atsruser

You seem to have answered the inverse question, AFAICS: how many vector potentials does a certain vector field admit?

Ooh yes...could be. But is the inverse of this even a question?

Original post by Gregorius

Ooh yes...could be. But is the inverse of this even a question?

It's a pretty boring one, I think, since differentiation isn't multivalued, but the OP answered my clarifying question with a yes. I've got a feeling that the setter or the OP posed the question incorrectly, and it's really meant to be the one that you asked.

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