When developing a physics curriculum, a major concern is the mathematical background of the student. The Physics2000 text was developed teaching premedical students who were supposed to have had one semester of calculus. Because many of the students had taken calculus several years previously, and had forgotten much of it, the physics text used strobe photographs and the computer to carefully introduce the calculus concepts such as velocity, acceleration, and the limiting process. By the time we got to electricity and magnetism in Part 2 of Physics2000 we relied on the student being familiar with the basic steps of differentiation and integration.
For students who have forgotten much of their calculus course, or those who have not had calculus but wish to study the Physics2000 text, we have written Chapter 1 of Calculus2000. This chapter not only covers all the calculus needed for the Physics2000 text, but is also carefully integrated with it. The chapter is much shorter than the typical introductory calculus text because the basic calculus concepts are discussed in the physics text and the calculus chapter only has to deal with the formalism.
After the introductory courses, the standard physics curriculum repeatedly goes over the same topics at successively higher mathematical levels. A typical example is the subject of electricity and magnetism which is taught using integral equations in the introductory course, using differential operators in an upper level undergraduate course, and then taught all over again in a graduate level course. In each of the courses it takes a while for the student to realize that this is just the same old subject dressed up in new math.
With Chapters 2 through 13 of the Calculus 2000, we introduce a different approach. We take the topics that we have already introduced in Physics2000, and show how these topics can be handled in progressively more sophisticated mathematical ways. Once we have introduced the mathematical concepts of gradient, divergence and curl in the calculus text, we can turn the integral form of Maxwell's equation into a wave equation for electric and magnetic fields. With the introduction of the Laplacian and complex variables, we can study Schrödinger's equation and begin to solve for the hydrogen wave patterns discussed in Chapter 38 of the physics text.
In Chapter 12 we discuss the concept of vorticity which is the curl of the velocity field. The focus is to develop an intuitive understanding of the nature of vorticity and the role it plays in fluid flows, particularly vortices and vortex rings.
Chapter 13 is an introduction to fluid dynamics. The idea is to bring our discussion of the velocity field up to the same level as our treatment of electric and magnetic fields. We begin with a derivation of the Navier-Stokes equation which applies to constant density viscous fluids. This is then converted into an equation for vortex dynamics from which we derive an extended form of the famous Helmholtz equation. We then use that to derive the well known properties of vortex motion such as the so-called Magnus force, and discuss the experiment Rayfield and Reif used to measure the circulation and core diameter of quantized vortices in superfluid helium.
Download the Calculus2000 text for free from the Downloads page.