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The present course on calculus of several variables is meant as a text, either for one semester following the First Course in Calculus, or for a
longer period if the calculus sequence is so structured.
In a one-semester course, I suggest covering most of the first part, omitting Chapter II, §3 and omitting some material from the chapter on
Taylor's formula in several variables, to suit the taste of the instructor and the class. One can then jump directly to the chapter on double and triple
integrals, which could in fact be treated immediately after Chapter I. If time allows, one can also cover the first section in the chapter on Green's theorem~ which gives a neat application of the techniques of double integrals and curve integrals .. Joining them in this fashion will make the student learn both techniques better for having used them in a significant context.
The first part has considerable unity of style. Essentially all the results are immediately corollaries of the chain rule. The main idea is that given a function of several variables, if we want to look at its values at two points P and Q, we join these points by a curve (often a straight line), and then look at the values of the function on that curve. By this device, we are able to reduce a large number of problems in several variables to problems and techniques in one variable. For instance, the directional derivative, the law of conservation of energy, and Taylor's formula, are handled in this manner.
I have included only that part of linear algebra which is immediately useful for the applications to calculus. My Introduction to Linear Algebra provides an appropriate text when a whole semester is devoted to the subject. Many courses are still structured to give primary emphasis to the analytic aspects, and only a few notions involving matrices and linear maps are needed to cover, say, the chain rule for mappings of one space into another, and to emphasize the importance of linear approximations.
The last chapter on surface integrals and Stokes' theorem could essentially be covered after Green's theorem and multiple integrals. The chapter
on the change of variables formula in multiple integration is the most expendable one, and can be omitted altogether without affecting the understanding of the rest of the book. Each instructor will adapt the material to the needs of any given class