[Book Cover]

Multivariable Calculus With Vectors, 1/e

Hartley Rogers, Massachusetts Institute of Technology

Published August, 1998 by Prentice Hall Engineering/Science/Mathematics

Copyright 1999, 789 pp.
ISBN 0-13-605643-1

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This text is for the third semester or fourth and fifth quarters of calculus; i.e., for multivariable or vector calculus courses. This text presents a conceptual underpinning for multivariable calculus that is as natural and intuitively simple as possible. More than its competitors, this book focuses on modeling physical phenomena, especially from physics and engineering, and on developing geometric intuition.


Greater emphasis is given to applications from physics and engineering.
Geometric intuition is particularly stressed. The synthetic, coordinate-free geometries of 2- and 3-dimensional Euclidean spaces (E^2 and E^3) have a primary role. These spaces, together with their assigned distance functions (“metrics”), are assumed as given and known structures. Basic definitions and properties are described, but a self-contained, logical treatment is not provided. For example, the Jordan curve theorem for piecewise-smooth curves in E^2 is stated and assumed but not proved. In accordance with this geometrical emphasis, scalar fields (mappings from a region of E^2 or E^3 to the real numbers) are given an equal place with functions of 2 and 3 variables.
Wherever possible, coordinate-free definitions are used. For example, the initial definitions of the differential operators divergence and curl and of the integral operators double integral and triple integral make no mention of a coordinate system. Furthermore, the physicist's {î,j,k} notation is used for vectors in a Cartesian coordinate system (instead of canonical-coordinate triples) to emphasize the subordinate status of any given coordinate system. When coordinates do appear in the text, they are seen as primarily useful for computational purposes.
New versions of certain calculus concepts are introduced in order to obtain a better fusion of mathematical argument with geometrical intuition. In particular, the central concept of limit is defined in terms of funnel functions. (In the terminology of mathematical logic, a funnel function is a convenient Skolem function associated with a statement of the form “for every …e>>0, there exists a …d>>0…”) Funnel functions are especially useful in multivariable calculus, where, for example, they give a simple and easily visualized definition of uniform continuity. Similarly, the central concept of definite integral is defined in terms of proper sequences of Riemann sums (a form of direct limit). For many students, this definition appears to be simpler and less artificial than the more familiar definition in terms of upper and lower sums. Finite additive measures and their derivatives are used to provide a more natural framework for the great theorems of vector integral calculus. The amount of emphasis given to these technical concepts will depend upon the level of rigor to which a student or teacher aspires. For many students, these technical concepts will remain plausible and instructive but peripheral features of their study.
In some cases, proofs are initially presented as plausibility arguments. Full proofs are then indicated in later portions of the text or in problem material.
The transition to Euclidean spaces of more than three dimensions occurs near the end of the text. The text's treatment of linear algebra in Chapter 33 leads to the emergence of the synthetic geometries of higher dimensional Euclidean spaces to which the geometric intuitions developed in three dimensions can be largely transferred. This seems preferable to the sacrifice of these geometric intuitions to a premature introduction of canonical coordinates.

Table of Contents

    1. Euclidean Geometry in Three Dimensions.
    2. Geometric Vectors and Vector Algebra.
    3. Vector Algebra with Cartesian Coordinates.
    4. Analytic Geometry in Three Dimensions.
    5. Calculus of One-Variable Vector Functions.
    6. Curves.
    7. Cylindrical Coordinates.
    8. Scalar Fields and Scalar Functions.
    9. Linear Approximation; the Gradient.
    10. The Chain Rule.
    11. Using the Chain Rule.
    12. Maximum-Minimum Problems.
    13. Constrained Maximum-Minimum Problems.
    14. Multiple Integrals.
    15. Iterated Integrals.
    16. Integrals in Polar, Cylindrical, or Spherical Coordinates.
    17. Curvilinear Coordinates and Change of Variables.
    18. Vector Fields.
    19. Line Integrals.
    20. Conservative Fields.
    21. Surfaces.
    22. Surface Integrals.
    23. Measures and Densities.
    24. Green's Theorem.
    25. The Divergence Theorem.
    26. Curl and Stoke's Theorem.
    27. Mathematical Applications.
    28. Physical Applications.
    29. Vectors and Matrices.
    30. Solving Simultaneous Equations by Row-reduction.
    31. Determinants.
    32. Matrix Algebra.
    33. Subspaces and Dimension.
    34. Topics in Linear Algebra.
    Answers to Selected Problems.


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