A ZERO Curl and a ZERO divergence

[Robt Massagli]

A ZERO Divergence Vector Field

There is theorem that is widely used in physics--e.g., electricity and magnetism for which I have no proof, yet we use this theorem at the drop of a hat. The theorem is this:

Given sufficient continuity and differentiability, every vector function A such that div(V) = 0 yields a vector function U such V = curl(U).

Is there a simple proof of this?

 

[algebrat]

I think a proof of this might go along the lines of constructing (a formula for) the vector potential, where the construction holds only for divergence-less vector fields.

Also, there are likely issues of the region over which you are integrating. Your statement above may only be guaranteed locally, a subtle point that is often glossed over in physics courses.

I'll try to describe this issue in the case of a curlless vector field being the gradient of some scalar potential.

The vector field (-y/r,x/r), I think that has zero curl, so locally it has a scalar potential, but globally it is not possible. The theorems don't apply because there is a singularity in the vector field at the origin.

You can visualize the potential function as a winding staircase, so the gradient points up the stairs, locally you can describe the height of the stairs, but if you make more than a full loop around the pole, your height will not be well defined over the plane.

 

[micromass]

This is not true. We need certain conditions on the domain in order to get this result. A nice condition that we can demand is that the domain of the vector field is open and star-shaped. For example, an open ball would satisfy this, or entire $\mathbb{R}^3$ would satisfy this as well.

In the case of an open and star-shaped domain, the result is true and is given by the so-called Poincaré lemma. Even more general, the Poincaré lemma holds for contractible domains.

A proof of this can be found in "Calculus on manifolds" by Spivak. The result is theorem 4-11 p94. However, it is stated in the language of differential forms. Exercise 4-19 in the same chapter relate differential forms to the more common notions of div, grad and curl.

 

[algebrat]

I made a typo above, I think my example should have been

(-y/r^2,x/r^2)

 

[Quantumjump]

There is theorem that is widely used in physics--e.g., electricity and magnetism for which I have no proof, yet we use this theorem at the drop of a hat. The theorem is this:

Given sufficient continuity and differentiability, every vector function A such that div(V) = 0 yields a vector function U such V = curl(U).

Is there a simple proof of this?

I think you are referrring to a particular case of a general result called Poincaré Lemma, which states under which condition a function called "potential" can exist, in a wide range of situations. It is indeed an extremely powerful result.