Applied Gradient Field Solutions

For gradient fields it is necessary to account for the Maxwell terms. That is, fields that exist in addition to the desired gradient of $B^{(0)}_z$ due to the fact that $\nabla^2 \phi_0 = 0$. Once $\phi_0$ is found, then $B^{(0)}$ is proportional to $\nabla \phi_0$.

For example, if an x-gradient in the z field is desired, $B^{(0)}_z = x$, then $\partial \phi_0 / \partial z = x$ (up to a constant). Consequently, $\phi_0 = xz$, which already satisfies $\nabla^2 \phi_0 = 0$. However, for $B^{(0)}_z = z$, then $\phi_0 = z^2/2$ cannot be used as $\nabla^2 \phi_0 = 1$. Consequently, $\phi_0 = z^2/2 - x^2/2$ (or some other equivalent using $x$ and $y$ in the second term) must be used instead.