Differential and Integral Calculus, Volume 1The classic introduction to the fundamentals of calculus Richard Courant's classic text Differential and Integral Calculus is an essential text for those preparing for a career in physics or applied math. Volume 1 introduces the foundational concepts of "function" and "limit", and offers detailed explanations that illustrate the "why" as well as the "how". Comprehensive coverage of the basics of integrals and differentials includes their applications as well as clearly-defined techniques and essential theorems. Multiple appendices provide supplementary explanation and author notes, as well as solutions and hints for all in-text problems. |
Contents
INTRODUCTORY REMARKS | 1 |
Further Examples of the Substitution Method | 3 |
The Continuum of Numbers | 7 |
The Concept of Function | 17 |
Functions of an Integral Variable Sequences of Numbers | 27 |
Further Discussion of the Concept of Limit | 38 |
The Concept of Limit where the Variable is Continuous | 46 |
14 | 52 |
Particle sliding down a Curve | 299 |
Properties of the Evolute | 307 |
CHAPTER VI | 315 |
Taylors Theorem | 322 |
Geometrical Applications | 331 |
CHAPTER VII | 342 |
Applications of the Mean Value Theorem and of Taylors Theorem | 349 |
Numerical Solution of Equations | 355 |
The Principle of the Point of Accumulation and its Applications | 58 |
Some Remarks on the Elementary Functions | 68 |
CHAPTER II | 76 |
The Derivative | 87 |
The Indefinite Integral the Primitive Function and the Funda mental Theorems of the Differential and Integral Calculus 88 | 109 |
Simple Methods of Graphical Integration | 119 |
Further Remarks on the Connexion between the Integral and the Derivative | 121 |
The Estimation of Integrals and the Mean Value Theorem of the Integral Calculus | 126 |
APPENDIX | 129 |
The Existence of the Definite Integral of a Continuous Function | 131 |
The Definite Integral | 133 |
The Relation between the Mean Value Theorem of the Differential Calculus and the Mean Value Theorem of the Integral Cal culus | 134 |
CHAPTER III | 136 |
The Corresponding Integral Formulæ 3 The Inverse Function and its Derivative | 144 |
Differentiation of a Function of a Function | 153 |
Maxima and Minima | 158 |
The Logarithm and the Exponential Function | 167 |
Some Applications of the Exponential Function | 178 |
The Hyperbolic Functions | 183 |
The Order of Magnitude of Functions | 189 |
Some Special Functions APPENDIX | 196 |
Remarks on the Differentiability of Functions | 199 |
Examples | 202 |
Some Special Formulæ | 203 |
CHAPTER IV | 204 |
Elementary Integrals | 205 |
The Method of Substitution CALCULUS | 211 |
Remarks on Functions which are not Integrable in Terms | 242 |
The Second Mean Value Theorem of the Integral Calculus | 256 |
Applications to the Theory of Plane Curves | 267 |
Examples | 290 |
APPENDIX | 361 |
Tests for Convergence and Divergence | 377 |
Sequences and Series of Functions | 383 |
Power Series | 398 |
Expansion of Given Functions in Power Series Method | 404 |
Power Series with Complex Terms | 410 |
Infinite Series and Improper Integrals | 417 |
APPENDIX | 455 |
Continuity | 463 |
The Chain Rule and the Differentiation of Inverse Functions | 472 |
Implicit Functions | 480 |
Multiple and Repeated Integrals | 486 |
CHAPTER XI | 501 |
The Nonhomogeneous Equation | 509 |
Additional Remarks on Differential Equations | 519 |
SUMMARY OF IMPORTANT THEOREMS AND FORMULE | 529 |
136 | 531 |
141 | 532 |
MISCELLANEOUS EXAMPLES | 549 |
144 | 562 |
153 | 563 |
167 | 564 |
178 | 565 |
183 | 566 |
196 | 567 |
207 | 568 |
ANSWERS AND HINTS | 571 |
INDEX | 577 |
| 611 | |
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Common terms and phrases
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