Physics with Calculus/Introduction/For Mathematicians
Math is a wonderful discipline. Too often though, mathematicians focus solely on theoretical mathematics, but this text can aid us in realizing the wonderful applications of math and, in this case, calculus. —Professor M.
Although physics is often thought of as an "applied mathematics" class, the relationship between physics and mathematics is far more interesting than just applying mathematics to the physical world. Often in the development of both physics and mathematics, the innovations in one area have driven new innovations in the other. One interesting example of this is the fact that when Newton wanted to test his theory of gravity he realized a major problem. His law of gravitation calculated the force between the two interacting objects as if all of the mass of the two objects was concentrated at a point at the center of the object. Newton was faced with a serious problem. How could he prove that all of the mass of an extended object could be truly considered to be at a point at the center of the object? This required a new kind of mathematics, and the integral calculus was devised. This illustrates the fact that, historically, the motivation for developing mathematics was often based on real problems. It is often not until later that all of the theoretical aspects of the mathematics are worked out. Although Newton is often credited with inventing the calculus, it is clear that he was very often more interested in developing mathematical models that would solve his scientific problems, and then proving his mathematical assumptions in a theoretical way. It was left to others that followed to put the calculus on the same solid foundation that had been developed for geometry and other branches of mathematics.
Innovations in mathematics can often drive new discoveries in science too. An interesting example of this the study of fractals. When they were first studied there was no practical use for them, but now physics and other sciences use them in the study of systems that are known to be chaotic.
So why would a mathematician want to study physics? It seems that one very powerful answer to this question is that it helps one to see into the minds of many of the great thinkers of the past. What motivated them? What where the challenges that they faced as they developed their ideas? They would often turn to mathematics as the language to express their ideas and solve their problems. The study of mathematics strengthens the tools of the scientist, and the study of science adds fullness to our understanding of mathematics.