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Vector CalculusMathematics
Multivariable CalculusMathematics

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, coordinatefree geometries of 2 and 3dimensional 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 selfcontained, logical treatment is not provided.
For example, the Jordan curve theorem for piecewisesmooth 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, coordinatefree 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 canonicalcoordinate
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.
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 OneVariable 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. MaximumMinimum Problems.
13. Constrained MaximumMinimum 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 Rowreduction.
31. Determinants.
32. Matrix Algebra.
33. Subspaces and Dimension.
34. Topics in Linear Algebra.
Answers to Selected Problems.
Index.
