Can I derive an equation for a vector field's divergence from its curl equation?

[Savant13]

Given an equation describing the curl of a vector field, is it possible to derive an equation for the originating vector field?

The divergence of the field is known to be zero at all points

 

[waht]

yea, you would have to solve partial differential equations:

$$\nabla \times \vec{F} = \vec{A}$$

$$\nabla \cdot \vec{F} = 0$$

 

[weichi]

There *are* expressions for this. You might be able to find them in something like Boas or Arfken and Weber.

If you are familiar with differential forms, many (most? all?) proofs of the converse of Poincare's lemma also give the expressions that you want. See, e.g, Flanders.

 

[Savant13]

There *are* expressions for this. You might be able to find them in something like Boas or Arfken and Weber.

If you are familiar with differential forms, many (most? all?) proofs of the converse of Poincare's lemma also give the expressions that you want. See, e.g, Flanders.

Where can I find this? I'm a high school student learning outside the classroom, so I don't have access to any resources that require subscription

 

[weichi]

More specific references:

"Mathematical Methods in the Physical Sciences" by Mary Boas. This is an undergrad level book that is great for covering math required to do physics. It has lots of problems, *many* with answers in the back. I think the only critical prereq is calculus, which you must have. This book was a big help to me while preparing for grad school.

I don't know whether Boas actually covers the specific question you have, but it's still a good book to take a look at! You can learn *a lot* from it.

"Mathematical Methods for Physicists" by Arfken and Weber. Graduate-level, covers much more than Boas. Again, don't know whether it covers your specific question. This is almost certainly going to be too difficult for a high-school student, but if you ever decide that you are comfortable with the material in Boas, you might want to peek in this book.

"Differential Forms with Applications to the Physical Sciences", by Harley Flanders. I like this book, but it does require some "mathematical maturity". I wouldn't really recommend looking at this yet.

But ... lo and behold, it's on google books, and the relevant passage is on pg 28-32. In fact, the example on pg 30 is exactly what you want! In the language of forms, Flanders' A, B, and C on this page are the x, y, and z components of your expression for curl B.

If these pages from Flanders interest you, you might want to look at threads here on physicsforums that discuss forms. I'm sure they have references that are easier to learn from.