Mathematical Description of the Hodograph Transformation

The basic goal is to derive a differential equation for $\psi$, the stream function, using the components of the velocity vector as the independent variables. In the hodograph plane, the final equation is linear, justifying the extra work of using it.

Most of this derivations below come directly from Emanuel; they are only summarized here. In the derivations, the shorthand suffix notation for a derivative is used: $\phi_x=\partial\phi/\partial x$.