Suppose A is a vector satisfying curlA = aA, a a function of position.

Is it possible to turn the volume integral of A^2 into a surface integral? I know I have to write A^2 as the divergence of something, but what?

Printable View

- October 7th 2010, 01:44 PMthefrogVolume integral to a surface integral ... divergence theorem
Suppose A is a vector satisfying curlA = aA, a a function of position.

Is it possible to turn the volume integral of A^2 into a surface integral? I know I have to write A^2 as the divergence of something, but what? - October 7th 2010, 05:31 PMAckbeet
A couple of questions for clarification.

1. What quantities here are vector quantities, and which ones are scalar?

2. Which multiplication do you mean when you write Is that Or is it

3. I'm also curious as to what physical quantity represents. Is that the vector magnetic potential? - October 7th 2010, 05:35 PMthefrog
Hi Ackbeet,

1/. a is scalar, A is a vector

2/. A^2 = A dot A

3/. Yes, it is a vector magnetic potential - October 7th 2010, 06:11 PMAckbeet
Ok, so re-writing your problem statement, I have the following:

Assume where is the vector magnetic potential defined by where is the magnetic field, and the scalar function depends only on the coordinates.

We wish to show that we can convert

into a surface integral via the divergence theorem.

The divergence theorem states that if is a continuously differentiable vector field defined on a neighborhood of then

Here's a question: do you know if is ever zero? Because, if not, you might be able to manipulate the equation

to simplify

I have another question: are you told that you can perform this transformation? Or are you merely trying to simplify a particular volume integral? If the second, can you provide a little more context?