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Scalar and Vector Calculus

To summarize what we've covered so far: Our study of electrodynamics is going to be founded on real and complex numbers that represent physical quantities with units, so we learned a bit about these kinds of (scalar) numbers. Since it is a kind of a map of what happens in space and time, we need to understand coordinates, vectors in a coordinate system, and various ways to multiply vectors. That led us to consider both tensor forms and coordinate transformation, as both of these will prove to be very useful if not essential. Coordinate transformations (at least) often form groups so we learned what a group was (and realized that we've been using e.g. the multiplication group all of our lives without realizing it.

It should come as no surprise that the remaining chunk of math we will need is calculus. After all, Newton invented calculus so he could invent physics, and electrodynamics is very much a part of physics. I'm not going to cover every single thing you learned in calculus classes in the past here (the chapter would be as long or longer than the entire book if I did) but rather will focus on showing you the path between the plain old calculus you already know (I profoundly hope) and the vector calculus you probably don't know anywhere near well enough unless you had a really extraordinary course in multivariate calculus and remember it all.

Let's begin pretty close to the beginning, with ordinary differentiation. Even here our treatment won't quite be ordinary, because we will not be reviewing this purely in the abstract. In all cases, where I refer to various (scalar and vector and possibly even tensor) functions, you should be thinking of those functions as numerically representing definite physical quantities, with units. The calculus we need is not abstract, it is descriptive, and it is this (possibly subtle) differentiation that separates the mathematician from the physicist.

Both a mathematician and a physicist may talk about doing things to or with a function $ f$ , but the physicist is always thinking about functions $ f$ that actually ``stand for something'' and $ f$ will usually be replaced by traditional symbols in application. To many mathematicians, $ f$ is just $ f$ - some function, any function - and it may or may not mean anything at all besides its own shape or form if even that is specified.



Subsections
next up previous contents
Next: Scalar Differentiation Up: Mathematical Physics Previous: The Inversion Group   Contents
Robert G. Brown 2017-07-11