Table Of Contents:
Sectionwise Dependence Tree xii

1 Review of Differential Calculus 1(48)

1.1 Vector Valued Functions 1(2)

1.2 Directional Derivatives and Total Derivative 3(10)

1.3 Linearity of the Derivative 13(5)

1.4 Inverse and Implicit Function Theorems 18(8)

1.5 Lagrange Multiplier Method 26(7)

1.6 Differentiability on Subsets of Euclidean Spaces 33(5)

1.7 Richness of Smooth Maps 38(7)

1.8 Miscellaneous Exercises for Chapter 1 45(4)

2 Integral Calculus 49(28)

2.1 Multivariable Integration 49(5)

2.2 Sard's Theorem 54(3)

2.3 Exterior Algebra 57(6)

2.4 Differential Forms 63(3)

2.5 Exterior Differentiation 66(3)

2.6 Integration on Singular Chains 69(6)

2.7 Miscellaneous Exercises for Chapter 2 75(2)

3 Submanifolds of Euclidean Spaces 77(24)

3.1 Basic Notions 77(3)

3.2 Manifolds with Boundary 80(3)

3.3 Tangent Space 83(4)

3.4 Special Types of Smooth Maps 87(6)

3.5 Transversality 93(2)

3.6 Homotopy and Stability 95(2)

3.7 Miscellaneous Exercises for Chapter 3 97(4)

4 Integration on Manifolds 101(20)

4.1 Orientation on Manifolds 101(5)

4.2 Differential Forms on Manifolds 106(1)

4.3 Integration on Manifolds 107(6)

4.4 De Rham Cohomology 113(7)

4.5 Miscellaneous Exercises for Chapter 4 120(1)

5 Abstract Manifolds 121(32)

5.1 Topological Manifolds 121(3)

5.2 Abstract Differential Manifolds 124(5)

5.3 Gluing Lemma 129(4)

5.4 Classification of 1-dimensional Manifolds 133(3)

5.5 Tangent Space and Tangent Bundle 136(5)

5.6 Tangents as Operators 141(4)

5.7 Whitney Embedding Theorems 145(5)

5.8 Miscellaneous Exercises for Chapter 5 150(3)

6 Isotopy 153(24)

6.1 Normal Bundle and Tubular Neighborhoods 153(5)

6.2 Orientation on Normal Bundle 158(2)

6.3 Vector Fields and Isotopies 160(9)

6.4 Patching-up Diffeomorphisms 169(5)

6.5 Miscellaneous Exercises for Chapter 6 174(3)

7 Intersection Theory 177(32)

7.1 Transverse Homotopy Theorem 177(2)

7.2 Oriented Intersection Number 179(2)

7.3 Degree of a Map 181(6)

7.4 Nonoriented Case 187(1)

7.5 Winding Number and Separation Theorem 188(4)

7.6 Borsuk-Ulam Theorem 192(2)

7.7 Hopf Degree Theorem 194(3)

7.8 Lefschetz Theory 197(8)

7.9 Some Applications 205(2)

7.10 Miscellaneous Exercises for Chapter 7 207(2)

8 Geometry of Manifolds 209(34)

8.1 Morse Functions 209(4)

8.2 Morse Lemma 213(4)

8.3 Operations on Manifolds 217(9)

8.4 Further Geometry of Morse Functions 226(8)

8.5 Classification of Compact Surfaces 234(9)

9 Lie Groups and Lie Algebras: The Basics 243(42)

9.1 Review of Some Matrix Theory 243(9)

9.2 Topological Groups 252(5)

9.3 Lie Groups 257(4)

9.4 Lie Algebras 261(4)

9.5 Canonical Coordinates 265(5)

9.6 Topological Invariance 270(1)

9.7 Closed Subgroups 271(1)

9.8 The Adjoint Action 272(2)

9.9 Existence of Lie Subgroups 274(5)

9.10 Foliation 279(6)
Hints/Solutions to Select Exercises 285(16)
Bibliography 301(4)
Index 305