Multivariable calculus with vectors


Language: Anglais

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676 p. · 23.5x17.8 cm · Hardback
This text is for the third term 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.
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.
  • 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 (C2 and E3) have a primary role.
  • Wherever possible, coordinate-free definitions are used. 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 some cases, proofs are initially presented as plausibility arguments. Full proofs are then indicated in later portions of the text of in problem material.