1 F=ma 1
1.1 Prelude to Newton's Principia 1
1.2 Equal Area in Equal Time 5
1.3 The Law of Gravity 9
1.4 Exercises 16
1.5 Reprise with Calculus 18
1.6 Exercises 26
2 Vector Algebra 29
2.1 Basic Notions 29
2.2 The Dot Product 34
2.3 The Cross Product 39
2.4 Using Vector Algebra 46
2.5 Exercises 50
3 Celestial Mechanics 53
3.1 The Calculus of Curves 53
3.2 Exercises 65
3.3 Orbital Mechanics 66
3.4 Exercises 75
4 Differential Forms 77
4.1 Some History 77
4.2 Differential 1-Forms 79
4.3 Exercises 86
4.4 Constant Differential 2-Forms 89
4.5 Exercises 96
4.6 Constant Differential к-Forms 99
4.7 Prospects 105
4.8 Exercises 107
5 Line Integrals,Multiple Integrals 111
5.1 The Riemann Integral 111
5.2 Line Integrals 113
5.3 Exercises 119
5.4 Multiple Integrals 120
5.5 Using Multiple Integrals 131
5.6 Exercises 134
6 Linear Transformations 139
6.1 Basic Notions 139
6.2 Determinants 146
6.3 History and Comments 157
6.4 Exercises 158
6.5 Invertibility 163
6.6 Exercises 169
7 Differential Calculus 171
7.1 Limits 171
7.2 Exercises 178
7.3 Directional Derivatives 181
7.4 The Derivative 187
7.5 Exercises 197
7.6 The Chain Rule 201
7.7 Using the Gradient 205
7.8 Exercises 207
8 Integration by Pullback 211
8.1 Change of Variables 211
8.2 Interlude with Lagrange 213
8.3 Exercises 216
8.4 The Surface Integral 221
8.5 Heat Flow 228
8.6 Exercises 230
9 Techniques of Differential Calculus 233
9.1 Implicit Differentiation 233
9.2 Invertibility 238
9.3 Exercises 244
9.4 Locating Extrema 248
9.5 Taylor's Formula in Several Variables 254
9.6 Exercises 262
9.7 Lagrange Multipliers 266
9.8 Exercises 277
10 The Fundamental Theorem of Calculus 279
10.1 Overview 279
10.2 Independence of Path 286
10.3 Exercises 294
10.4 The Divergence Theorems 297
10.5 Exercises 310
10.6 Stokes'Theorem 314
10.7 Summary for R3 321
10.8 Exercises 323
10.9 Potential Theory 326
11 E=mc2 333
11.1 Prelude to Maxwell's Dynamical Theory 333
11.2 Flow in Space-Time 338
11.3 Electromagnetic Potential 345
11.4 Exercises 349
11.5 Special Relativity 352
11.6 Exercises 360
Appendices 361
A An Opportunity Missed 361
B Bibliography 365
C Clues and Solutions 367
Index 382