Vector and Scalar potential conditions

Question

Given that A is a 3-by-3 matrix, with constants a11, a12, a13, ..., a31, a32, a33. And vector F is A*x, where x is a vector. What are the general conditions on vector A for vector F to have a 1) scalar potential and 2) vector potential?

Answer 1

I'm going to use the Einstein summation convention in my answer because I find it easier to work with for these kinds of problems.

We have $$ F^i = a^i_j x^j $$ So $$ \nabla \cdot F = \frac{\partial F^i}{\partial x^i}= \frac{\partial}{\partial x^i} \left(a^i_j x^j\right) = a^i_j \frac{\partial x^j}{\partial x^i} = a^i_j \delta^j_i = a^i_i = \mathrm{Tr} \ A $$ We also find $$ (\nabla \times F)^i = \epsilon_{ijk}\frac{\partial F^k}{\partial x^j} = \epsilon_{ijk} \frac{\partial}{\partial x^j} \left(a^k_l x^l\right) = \epsilon_{ijk} a^k_l \delta^l_j = \epsilon_{ijk} a^k_j $$

Working with the usual assumptions, a vector field has a scalar potential if it is curl free. This means $$ 0 = (\nabla \times F)^i = \epsilon_{ijk}a^k_j $$ which you can check means that $A$ is symmetric.

A vector field has a vector potential if $A$ is divergence free. This means $$ 0 = \nabla \cdot F = \mathrm{Tr} \ A $$ i.e. $A$ is traceless.