Problem 12.9 in the infamous Jackson Classical Electrodynamics asks you to find (or prove) that magnetic field lines of the Earth are described by the equation

Your starting point is the equation for the magnetic field of a dipole,

Sure, you can look at a vector plot of the field, and join the arrows to make lines. This is the kind of thing they have us do in introductory E&M classes. But how do you do it mathematically? There are two ways that I know of:

1. Define some scalar function \(f(\vec{r})\) whose gradient is everywhere perpendicular to the magnetic field, by the condition \(\vec{B}\cdot\grad f = 0\), and use that condition to solve for \(\grad f\). This is useful because the level curves of a function (lines of constant \(f\)) are perpendicular to the gradient. So once you find what the gradient is, you can integrate it to find \(f(\vec{r})\), and get the level curves from that.
2. Or, just use the traditional “rise over run” definition of slope. Imagine a spherical polar coordinate grid …