Calculus Two Book

	Remembrance of Things Past	1
0.2	Sets	1
0.3	Functions	3
1	The Algebra of R[superscript n]	5
1.1	The Space R[superscript 2]	7
1.2	The Space R[superscript n]	12
1.3	Linear Subspaces	16
1.4	The Linear Subspaces of R[superscript 3]	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[superscript n]	71
2.1	The Norm of a Vector	71
2.2	The Inner Product	75
2.3	Hyperplanes and Orthogonality in R[superscript n]	89
2.4	The Cross Product in R[superscript 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[superscript n] [actual symbol not reproducible] R[superscript n]	167
3.8	Isometries	187
	Extra: Linear Maps on Function Spaces	202
	Extra: Linear Programming	204
4	Curves: Mappings F:R [actual symbol not reproducible] R[superscript q]	207
4.1	Limits, Continuity, and Curves	210
4.2	The Tangent Map	225
4.3	Are 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[superscript n]	302
6	Functions f:R[superscript n] [actual symbol not reproducible] 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[superscript n] [actual symbol not reproducible] R[superscript q]	417
8.1	Affine Approximation and Tangency	423
8.2	Rules for Calculating Derivatives	430
8.3	Surfaces in R[superscript q]	438
8.4	Vector Fields	448
9	Integration in R[superscript 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
	Answers and Partial Answers to Selected Exercises	581
	Index	611