Vector Fields and the Unity of Mathematics and Physics

We give an argument that magnetic monopoles should not exist. It is based on the concept of the index of a vector field. The thrust of the argument is that indices of vector fields are invariants of space-time orientation and of coordinate changes, and thus physical vector fields should preserve indices. The index is defined inductively by means of an equation called the Law of Vector Fields. We give extended philosophical arguments that this Law of Vector Fields should play an important role in mathematics, and we back up this contention by using it in a mechanical way to greatly generalize the Gauss--Bonnet theorem and the Brouwer fixed point theorem and get new proofs of many other theorems. We also give some other suggestions for using the Law and index in physics.