iv Contents
4.3. The Mean Value Theorem 89
4.4. L’Hˆ opital’s Rule 93
Chapter 5. The Integral 101
5.1. Definition of the Integral 101
5.2. Existence and Properties of the Integral 108
5.3. The Fundamental Theorems of Calculus 114
5.4. Logs, Exponentials, Improper Integrals 120
Chapter 6. Infinite Series 129
6.1. Convergence of Infinite Series 129
6.2. Tests for Convergence 134
6.3. Absolute and Conditional Convergence 140
6.4. Power Series 146
6.5. Taylor’s Formula 153
Chapter 7. Convergence in Euclidean Space 161
7.1. Euclidean Space 161
7.2. Convergent Sequences of Vectors 168
7.3. Open and Closed Sets 174
7.4. Compact Sets 179
7.5. Connected Sets 184
Chapter 8. Functions on Euclidean Space 191
8.1. Continuous Functions of Several Variables 191
8.2. Properties of Continuous Functions 197
8.3. Sequences of Functions 202
8.4. Linear Functions, Matrices 207
8.5. Dimension, Rank, Lines, and Planes 215
Chapter 9. Differentiation in Several Variables 223
9.1. Partial Derivatives 223
9.2. The Differential 229
9.3. The Chain Rule 236
9.4. Applications of the Chain Rule 242
9.5. Taylor’s Formula 251
9.6. The Inverse Function Theorem 260
9.7. The Implicit Function Theorem 266
Chapter 10. Integration in Several Variables 275
10.1. Integration over a Rectangle 275
10.2. Jordan Regions 282
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