Friday, April 3, 2009

Div, Grad, Curl, and All That

Russ Hobbie and I assume that readers of the 4th edition of Intermediate Physics for Medicine and Biology know the basics of calculus (our preface states that “calculus is used without apology”). We even introduce some concepts from vector calculus, such as the divergence, gradient, and curl. Although these vector derivatives are crucial for understanding topics such as diffusion and electricity, many readers may be unfamiliar with them. These functions are even more complicated in curvilinear coordinate systems, and in Appendix L we summarize how to write the divergence, gradient, curl, and Laplacian in rectangular, cylindrical, and spherical coordinates.

Div, Grad, Curl, and All That,  by H. M. Schey, superimposed on Intermediate Physics for Medicine and Biology.
Div, Grad, Curl, and All That,
by H. M. Schey.
When I was a young physics student at the University of Kansas, Dr. Jack Culvahouse gave me a book that helped explain vector calculus: Div, Grad, Curl, and All That: An Informal Text on Vector Calculus, by H. M. Schey. For me, this book made clear and intuitive what had been confusing and complicated. By defining the divergence and curl in terms of surface and line integrals, I suddenly could understand what these seemingly random collections of partial derivatives meant. One can hardly make sense of Maxwell’s equations of electromagnetism without vector calculus (try reading a textbook from Maxwell’s era before vector calculus was invented if you don't believe me). In fact, Schey introduces vector calculus using electromagnetism as his primary example:
In this text the subject of vector calculus is presented in the context of simple electrostatics. We follow this procedure for two reasons. First, much of vector calculus was invented for use in electromagnetic theory and is ideally suited to it. This presentation will therefore show what vector calculus is and at the same time give you an idea of what it's for. Second, we have a deep-seated conviction that mathematics—in any case some mathematicsis best discussed in a context that is not exclusively mathematical. Thus, we will soft-pedal mathematical rigor, which we think is an obstacle to learning this subject on a first exposure to it, and appeal as much as possible to physical and geometric intuition.
For readers of Intermediate Physics for Medicine and Biology who get stuck when we delve into vector calculus, I suggest setting our book aside for a few days (but only a few!) to read Div, Grad, Curl, and All That. Not only will you be able to understand our book better, but youll find this background useful in many other fields of physics, math, and engineering.

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