Calculus Two by Flanigan and Kazdan

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

• 0.2 Sets 1
• 0.3 Functions 3

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