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Latest revision as of 14:51, 23 November 2011
Calculus Two. Linear and Nonlinear Functions by F.J. Flanigan and J.L. Kazdan.
Calculus Two might mean "Calculus 2.0" or simply "advanced calculus".
Used it twice, for Vector calculus: course. A fine book but I'd prefer something more "compact".
Table of Contents
Remembrance of Things Past 1
1 The Algebra of $R^n$ 5
- 1.1 The space $R^2$ 7
- 1.2 The space $R^n$ 12
- 1.3 Linear subspaces 16
- 1.4 The linear subspaces of $R^n$ 22
- 1.5 Systems of equations 32
- 1.6 Affine subspaces 49
- 1.7 The Dimension of a vector space 58
- Extra: Function spaces 68
2 The geometry of $R^n$ 71
- 2.1 The Norm of a vector 71
- 2.2 The Inner product 75
- 2.3 Hyperplanes and orthogonality in $R^n$ 89
- 2.4 The Cross product in $R^3$ 92
- Extra: Euclid using Vectors 99
3 Linear Functions 103
- 3.1 Definition and Basic Properties 104
- 3.2 Linear maps and Linear Subspaces 116
- 3.3 A Special Case: Linear functionals 133
- 3.4 The Algebra of Linear Maps 135
- 3.5 Matrices 143
- 3.6 Affine maps 161
- 3.7 Another Special Case: $L:R^n \rightarrow R^n$ 167
- 3.8 Isometries 187
- Extra: Linear Maps on Function Spaces 202
- Extra: Linear Programming 204
4 Curves: Mappings $F:R \rightarrow R^q$ 207
- 4.1 Limits, Continuity, and Curves 210
- 4.2 The Tangent Map 225
- 4.3 Arc length and Curvature 244
5 Topics for Review and Preview 263
- 5.1 Further Concepts and Problems 263
- 5.2 Some Challenging Problems 285
- 5.3 Gravitational Motion 292
- 5.4 Geometry in $R^n$ 302
6 Functions $f:R^n \rightarrow R$ 313
- 6.1 Continuity and Limits 319
- 6.2 Directional derivatives 333
- 6.3 Partial derivatives 339
- 6.4 Tangency and Affine approximation 348
- 6.5 The Main Theorems 357
- 6.6 The World of First Derivatives 368
7 Scalar-Valued Functions and Extrema 369
- 7.1 Local extrema are Critical points 372
- 7.2 The Second derivative 379
- 7.3 The Second derivative Test 388
- 7.4 Global extrema 401
- 7.5 Constrained extrema 405
8 Vector Functions $F:R^n \rightarrow R^q$ 417
- 8.1 Affine approximation and Tangency 423
- 8.2 Rules for Calculating Derivatives 430
- 8.3 Surfaces in $R^q$ 438
- 8.4 Vector fields 448
9 Integration in $R^n$ 455
- 9.1 Estimating the Value of Integrals 458
- 9.2 Computing Integrals Exactly 473
- 9.3 Theory of the Integral 494
- 9.4 Change of variables 500
- 9.5 Surface area 523
10 Vector integrals and Stokes' theorem 529
- 10.1 Line integrals 530
- 10.2 Stokes' Theorem in the Plane 543
- 10.3 Surface integrals 554
- 10.4 Independence of path: Potential Functions 570