Synopses & Reviews
This is a new, revised edition of this widely known text. All of the basic topics in calculus of several variables are covered, including vectors, curves, functions of several variables, gradient, tangent plane, maxima and minima, potential functions, curve integrals, Green's theorem, multiple integrals, surface integrals, Stokes' theorem, and the inverse mapping theorem and its consequences. The presentation is self-contained, assuming only a knowledge of basic calculus in one variable. Many completely worked-out problems have been included.
The present course on calculus of several variables is meant as a text, either for one semester following A First Course in Calculus, or for a year if the calculus sequence is so structured. For a one-semester course, no matter what, one should cover the first four chapters, up to the law of conservation of energy, which provides a beautiful application of the chain rule in a physical context, and ties up the mathematics of this course with standard material from courses on physics. Then there are roughly two possibilities: One is to cover Chapters V and VI on maxima and minima, quadratic forms, critical points, and Taylor's formula. One can then finish with Chapter IX on double integration to round off the one-term course. The other is to go into curve integrals, double integration, and Green's theorem, that is Chapters VII, VIII, IX, and X, 1. This forms a coherent whole."
This new, revised edition covers all of the basic topics in calculus of several variables, including vectors, curves, functions of several variables, gradient, tangent plane, maxima and minima, potential functions, curve integrals, Green's theorem, multiple integrals, surface integrals, Stokes' theorem, and the inverse mapping theorem and its consequences. It includes many completely worked-out problems.
Table of Contents
I: Basic Material. 1: Vectors. 2: Differentiation of Vectors. 3: Functions of Several Variables. 4: The Chain Rule and the Gradient. II: Maxima, Minima, and Taylor's Formula. 5: Maximum and Minimum. 6: Higher Derivatives. III: Curve Integrals and Double Integrals. 7: Potential Functions. 8: Curve Integrals. 9: Double Integrals. 10: Green's Theorem. IV: Triple and Surface Integrals. 12: Triple Integrals. V: Mappings, Inverse Mappings, and Change of Variables Formula. 13: Matrices. 14: Linear Mappings. 15: Determinants. 16: Applications to Functions of Several Variables. 17: The Change of Variables Formula. Appendix: Fourier Series.