This last SparkNote concerning magnetic fields is a purely theoretical one. We don't examine particular configurations of wires, solenoids, and magnets. We don't look at the force on moving charges. Instead, we simply look at magnetic fields as a special kind of vector field, and describe them purely in terms of the mathematical properties of such a field. We are able to do this such that a magnetic field can be described completely by two simple equations. In essence, we are able to compress all of the preceding topics into two equations.
Before we make these mathematical statements, however, we must first develop the multivariable calculus used to derive our equations. We develop the concepts of divergence and curl, and introduce the two important theorems: Stokes' Theorem and Gauss' Theorem. Equipped with this background we can then apply the math to magnetic fields, generating our two important equations.
By finally analyzing magnetic fields on a purely theoretical level we complete our study of magnetic fields. We have looked at the effects of magnetic fields, the sources of magnetic fields and, finally, in this SparkNote, the theory of magnetic fields. This complex topic must be attacked from many angles in order for us to understand it.