And so we find that the potential function is constant. Since the vector field is only a two-dimensional vector field, we can check the simpler relations to determine whether it is conservative. We see that, letting , Since these partial derivatives are not equal, we know that is not a gradient field and therefore has no potential. So Now we compute the partial derivatives. For simplicity, I show only the first term, .
Similar terms arise from the other derivatives.
In the end, you will find that div , as long as . Because the divergence of this vector field is undefined at the origin, we cannot use the divergence theorem directly to compute the flux through a cube centered at the origin. However, we can break the region of integration up into two pieces. Since the vector field is pointing radially outward, it will be easy to calculate the flux through a sphere centered at the origin. For this reason, we break the cube up into two parts.