I’ve always thought that math physics have had a very intimate relationship. After all, it seems as if one could never get very far in studying physics without a firm grasp on a few mathematical concepts. What I would argue is most basic is an understanding algebra. Many people view physics as that hard course that they needed to take in college for some reason unbeknownst to them. To most, this course consists of a large set of random equations. The task of those taking the course then becomes figuring out what equations to use for which problem. To do this, one definitely needs to be comfortable manipulating and rearranging equations.
As one progresses in their physics career, math becomes more of a necessity. It seems like math is the language of physics. A prime example can be seen when studying electrodynamics. One can have a large set of equations which apply only to specific situations. But with a little knowledge of vector calculus one can greatly reduce the number of equations needed to four, namely Maxwell’s equations. Though he discovered none of them and only “fixed” one, his name is associated with these equations because he was one of the first people to recognize how closely they were related. For those of you who aren’t familiar, Maxwell’s equations are as follows:
Where E denotes electric vector field, B denotes magnetic vector field, ρ denotes the electric charge density, J denotes electric current density, and μ (naught) is the permeability of free space, and ε (naught) is the permittivity of free space. Let’s just examine the first equation here. First let’s make sure everyone knows what the del operator (the “upside down” triangle). It can be thought of a differential vector. In three dimensions it is
When this operator is multiplied by a vector field via scalar product, as it is in the first of Maxwell’s equations, it is called divergence. In loose terms, it gives a measure of how much the vector field is expanding from that point. If we are to integrate both sides of the first of Maxwell’s equations we are able to apply the divergence theorem on the left hand side. The divergence theorem states that the integral of the divergence of a vector field over some volume is equal to the surface integral of the flux of the vector field over the surface enclosing the volume. More clearly stated:
After integrating, the left hand side of the equation becomes the integral of the flux over a surface integral. The right hand side becomes the charge enclosed by the surface, divided by the permittivity constant. But what does this mean? It means that the amount of electric flux through a surface is directly related to the amount of charge enclosed in that surface. What if we extend our surface without enclosing any more charge? Assuming there’s some good symmetry (usually a sphere), we can deduce that the strength of the electric field decreases as we move away from the source (the charge enclosed). This may seem like a very round about way to say what could be said in a sentence or two. This is true to an extent, but there is no way one could conclude anything quantitative from a couple of sentences. Thus the mathematical representation is much more robust than saying a few words about how electric fields behave.
I’d say that’s enough of that, so lets get back to the question at hand. How exactly are math and physics related? Is it true that math is merely a tool on the belt of a physicist? If this were true, I think the relationship could be classified as being communal, meaning that physics is the only beneficiary. One needs only to come up with a counter example to disprove this claim. The perfect counter example can be found in one of my favorite mathematical stories, and perhaps one of the best known. Newton’s work on the development of differential calculus is an obvious way in which physics has helped to advance mathematics (I know Leibniz played his role as well, but let’s not get into that). Newton made leaps and bounds towards the calculus we know today, but was no mathematician. He didn’t prove what he know worked, as is the case for many physicists. He did however lay down the ground work.
I’m sure there are numerous other examples where physics has lent ideas to mathematics and vice versa, but I’m running out time to write this post. What we can take away though is that math and physics are intimately related in a mutual relationship. Both help to advance the other. I feel that this should be kept in mind when teaching the two subjects. This allows for students to see how close the two subjects are related. It also will prepare them to make the same types of advancements that have been made throughout history. If we teach math and physics as separate entities, in the future we might treat them as so, which could hinder progress in both fields.