Vector Potential: How to Find it?

[PrinceOfDarkness]

Vector V=x^2i+3xz^2j-2xzk
The divergence of this vector is zero. So it can be expressed as the curl of a vector. I have to find that vector, which is also called the vector potential.
But I don't know how to find it. When I have to find the scalar potential, then it is easier to equate components and then integrate. But in this case, if I equate components, I get two variables of integration. How can I integrate then? Is it possible to find a vector potential like this?

 

[CarlB]

But in this case, if I equate components, I get two variables of integration. How can I integrate then? Is it possible to find a vector potential like this?

It sounds like you're doing the right thing. What you're getting is what the physicists call a "choice of gauge" for the potential.

For the scalar potential, there was only one degree of freedom, and you know that that amounted to just an aribitrary offset. That is, if you add a constant to the potential, the potential still gives the same force.

For the vector potential, there are more than one degree of freedom, and this means that instead of just adding a constant, you can actually add a function of position. The function must be one that has a gradient of zero, but there are plenty of those.

And so, when you look for the vector potential, you're going to get more constants.

Carl