Differential and integral calculus. Vol. I / by R. Courant ; translated by E. J. McShane.
Idioma: Inglés Lenguaje original: Alemán Editor: London : Blackie & Son, 1937Edición: 2nd edDescripción: xiii, 616 p. ; 23 cmTítulos uniformes: Vorlesungen über Differential- und Integralrechnung. Inglés Otra clasificación: 26-01Chapter I INTRODUCTION 1. The Continuum of Numbers [5] 2. The Concept of Function [14] 3. More Detailed Study of the Elementary Functions [22] 4. Functions of an Integral Variable. Sequences of Numbers [27] 5. The Concept of the Limit of a Sequence [29] 6. Further Discussion of the Concept of Limit [38] 7. The Concept of Limit where the Variable is Continuous [46] 8. The Concept of Continuity [49] APPENDIX I Preliminary Remarks [56] 1. The Principle of the Point of Accumulation and its Applications [58] 2. Theorems on Continuous Functions [63] 3. Some Remarks on the Elementary Functions [68] APPENDIX II 1. Polar Co-ordinates [71] 2. Remarks on Complex Numbers [73] Chapter II THE FUNDAMENTAL IDEAS OF THE INTEGRAL AND DIFFERENTIAL CALCULUS 1. The Definite Integral [76] 2. Examples [82] 3. The Derivative [88] 4. The Indefinite Integral, the Primitive Function, and the Fundamental Theorems of the Differential and Integral Calculus [109] 5. Simple Methods of Graphical Integration [119] 6. Further Remarks on the Connexion between the Integral and the Derivative [121] 7. The Estimation of Integrals and the Mean Value Theorem of the Integral Calculus [126] APPENDIX 1. The Existence of the Definite Integral of a Continuous Function [131] 2. The Relation between the Mean Value Theorem of the Differential Calculus and the Mean Value Theorem of the Integral Calculus [134] Chapter III DIFFERENTIATION AND INTEGRATION OF THE ELEMENTARY FUNCTIONS 1. The Simplest Rules for Differentiation and their Applications [136] 2. The Corresponding Integral Formulae [141] 3. The Inverse Function and its Derivative [144] 4. Differentiation of a Function of a Function [153] 5. Maxima and Minima [158] 6. The Logarithm and the Exponential Function [167] 7. Some Applications of the Exponential Function [178] 8. The Hyperbolic Functions [183] 9. The Order of Magnitude of Functions [189] APPENDIX 1. Some Special Functions [196] 2. Remarks on the Differentiability of Functions [199] 3. Some Special Formulae [201] Chapter IV FURTHER DEVELOPMENT OF THE INTEGRAL CALCULUS 1. Elementary Integrals [205] 2. The Method of Substitution [207] 3. Further Examples of the Substitution Method [214] 4. Integration by Parts [218] 5. Integration of Rational Functions [226] 6. Integration of Some Other Classes of Functions [234] 7. Remarks on Functions which are not Integrable in Terms of Elementary Functions [242] 8. Extension of the Concept of Integral. Improper Integrals [245] APPENDIX The Second Mean Value Theorem of the Integral Calculus [256] Chapter V APPLICATIONS 1. Representation of Curves [258] 2. Applications to the Theory of Plane Curves [267] 3. Examples [287] 4. Some very Simple Problems in the Mechanics of a Particle [292] 5. Further Applications: Particle sliding down a Curve [299] 6. Work [304] APPENDIX 1. Properties of the Evolute [307] 2. Areas bounded by Closed Curves [311] Chapter VI TAYLOR’S THEOREM AND THE APPROXIMATE EXPRESSION OF FUNCTIONS BY POLYNOMIALS 1. The Logarithm and the Inverse Tangent [315] 2. Taylor’s Theorem [320] 3. Applications. Expansions of the Elementary Functions [326] 4. Geometrical Applications [331] APPENDIX 1. Example of a Function which cannot be expanded in a Taylor Series [336] 2. Proof that e is Irrational [336] 3. Proof that the Binomial Series Converges [337] 4. Zeros and Infinities of Functions, and So-called Indeterminate Expressions [333] Chapter VII NUMERICAL METHODS Preliminary Remarks [342] 1. Numerical Integration [342] 2. Applications of the Mean Value Theorem and of Taylor’s Theorem. The Calculus of Errors [349] 3. Numerical Solution of Equations [355] APPENDIX Stirling’s Formula [361] Chapter VIII INFINITE SERIES AND OTHER LIMITING PROCESSES Preliminary Remarks [365] 1. The Concepts of Convergence and Divergence [366] 2. Tests for Convergence and Divergence [377] 3. Sequences and Series of Functions [383] 4. Uniform and Non-uniform Convergence [386] 5. Power Series [398] 6. Expansion of Given Functions in Power Series. Method of Undetermined Coefficients. Examples [404] 7. Power Series with Complex Terms [410] APPENDIX 1. Multiplication and Division of Series [415] 2. Infinite Series and Improper Integrals [417] 3. Infinite Products [419] 4. Series involving Bernoulli’s Numbers [422] Chapter IX FOURIER SERIES 1. Periodic Functions [425] 2. Use of Complex Notation [433] 3. Fourier Series [437] 4. Examples of Fourier Series [440] 5. The Convergence of Fourier Series [447] APPENDIX Integration of Fourier Series [455] Chapter X A SKETCH OF THE THEORY OF FUNCTIONS OF SEVERAL VARIABLES 1. The Concept of Function in the Case of Several Variables [458] 2. Continuity [463] 3. The Derivatives of a Function of Several Variables [466] 4. The Chain Rule and the Differentiation of Inverse Functions [472] 5. Implicit Functions [480] 6. Multiple and Repeated Integrals [486] Chapter XI THE DIFFERENTIAL EQUATIONS FOR THE SIMPLEST TYPES OF VIBRATION 1. Vibration Problems of Mechanics and Physics [502] 2. Solution of the Homogeneous Equation. Free Oscillations [504] 3. The Non-homogeneous Equation. Forced Oscillations [509] 4. Additional Remarks on Differential Equations [519] Summary of Important Theorems and Formulae [529] Miscellaneous Examples [549] Answers and Hints [571] Index [611]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
|---|---|---|---|---|---|---|---|---|
Libros
|
Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 26 C858-2 (Browse shelf) | Vol. I | Available | A-2946 |
Traducción de: Vorlesungen über Differential- und Integralrechnung.
Esta edición solo consta del vol. I; para el vol. II véase la 1ra edición.
Chapter I --
INTRODUCTION --
1. The Continuum of Numbers [5] --
2. The Concept of Function [14] --
3. More Detailed Study of the Elementary Functions [22] --
4. Functions of an Integral Variable. Sequences of Numbers [27] --
5. The Concept of the Limit of a Sequence [29] --
6. Further Discussion of the Concept of Limit [38] --
7. The Concept of Limit where the Variable is Continuous [46] --
8. The Concept of Continuity [49] --
APPENDIX I --
Preliminary Remarks [56] --
1. The Principle of the Point of Accumulation and its Applications [58] --
2. Theorems on Continuous Functions [63] --
3. Some Remarks on the Elementary Functions [68] --
APPENDIX II --
1. Polar Co-ordinates [71] --
2. Remarks on Complex Numbers [73] --
Chapter II --
THE FUNDAMENTAL IDEAS OF THE INTEGRAL --
AND DIFFERENTIAL CALCULUS --
1. The Definite Integral [76] --
2. Examples [82] --
3. The Derivative [88] --
4. The Indefinite Integral, the Primitive Function, and the Fundamental Theorems of the Differential and Integral Calculus [109] --
5. Simple Methods of Graphical Integration [119] --
6. Further Remarks on the Connexion between the Integral and the Derivative [121] --
7. The Estimation of Integrals and the Mean Value Theorem of the Integral Calculus [126] --
APPENDIX --
1. The Existence of the Definite Integral of a Continuous Function [131] --
2. The Relation between the Mean Value Theorem of the Differential Calculus and the Mean Value Theorem of the Integral Calculus [134] --
Chapter III --
DIFFERENTIATION AND INTEGRATION OF THE ELEMENTARY FUNCTIONS --
1. The Simplest Rules for Differentiation and their Applications [136] --
2. The Corresponding Integral Formulae [141] --
3. The Inverse Function and its Derivative [144] --
4. Differentiation of a Function of a Function [153] --
5. Maxima and Minima [158] --
6. The Logarithm and the Exponential Function [167] --
7. Some Applications of the Exponential Function [178] --
8. The Hyperbolic Functions [183] --
9. The Order of Magnitude of Functions [189] --
APPENDIX --
1. Some Special Functions [196] --
2. Remarks on the Differentiability of Functions [199] --
3. Some Special Formulae [201] --
Chapter IV --
FURTHER DEVELOPMENT OF THE INTEGRAL CALCULUS --
1. Elementary Integrals [205] --
2. The Method of Substitution [207] --
3. Further Examples of the Substitution Method [214] --
4. Integration by Parts [218] --
5. Integration of Rational Functions [226] --
6. Integration of Some Other Classes of Functions [234] --
7. Remarks on Functions which are not Integrable in Terms of Elementary Functions [242] --
8. Extension of the Concept of Integral. Improper Integrals [245] --
APPENDIX --
The Second Mean Value Theorem of the Integral Calculus [256] --
Chapter V --
APPLICATIONS --
1. Representation of Curves [258] --
2. Applications to the Theory of Plane Curves [267] --
3. Examples [287] --
4. Some very Simple Problems in the Mechanics of a Particle [292] --
5. Further Applications: Particle sliding down a Curve [299] --
6. Work [304] --
APPENDIX --
1. Properties of the Evolute [307] --
2. Areas bounded by Closed Curves [311] --
Chapter VI --
TAYLOR’S THEOREM AND THE APPROXIMATE EXPRESSION OF FUNCTIONS BY POLYNOMIALS --
1. The Logarithm and the Inverse Tangent [315] --
2. Taylor’s Theorem [320] --
3. Applications. Expansions of the Elementary Functions [326] --
4. Geometrical Applications [331] --
APPENDIX --
1. Example of a Function which cannot be expanded in a Taylor Series [336] --
2. Proof that e is Irrational [336] --
3. Proof that the Binomial Series Converges [337] --
4. Zeros and Infinities of Functions, and So-called Indeterminate Expressions [333] --
Chapter VII --
NUMERICAL METHODS --
Preliminary Remarks [342] --
1. Numerical Integration [342] --
2. Applications of the Mean Value Theorem and of Taylor’s Theorem. The Calculus of Errors [349] --
3. Numerical Solution of Equations [355] --
APPENDIX --
Stirling’s Formula [361] --
Chapter VIII --
INFINITE SERIES AND OTHER LIMITING PROCESSES --
Preliminary Remarks [365] --
1. The Concepts of Convergence and Divergence [366] --
2. Tests for Convergence and Divergence [377] --
3. Sequences and Series of Functions [383] --
4. Uniform and Non-uniform Convergence [386] --
5. Power Series [398] --
6. Expansion of Given Functions in Power Series. Method of Undetermined Coefficients. Examples [404] --
7. Power Series with Complex Terms [410] --
APPENDIX --
1. Multiplication and Division of Series [415] --
2. Infinite Series and Improper Integrals [417] --
3. Infinite Products [419] --
4. Series involving Bernoulli’s Numbers [422] --
Chapter IX --
FOURIER SERIES --
1. Periodic Functions [425] --
2. Use of Complex Notation [433] --
3. Fourier Series [437] --
4. Examples of Fourier Series [440] --
5. The Convergence of Fourier Series [447] --
APPENDIX --
Integration of Fourier Series [455] --
Chapter X --
A SKETCH OF THE THEORY OF FUNCTIONS OF SEVERAL VARIABLES --
1. The Concept of Function in the Case of Several Variables [458] --
2. Continuity [463] --
3. The Derivatives of a Function of Several Variables [466] --
4. The Chain Rule and the Differentiation of Inverse Functions [472] --
5. Implicit Functions [480] --
6. Multiple and Repeated Integrals [486] --
Chapter XI --
THE DIFFERENTIAL EQUATIONS FOR THE SIMPLEST TYPES OF VIBRATION --
1. Vibration Problems of Mechanics and Physics [502] --
2. Solution of the Homogeneous Equation. Free Oscillations [504] --
3. The Non-homogeneous Equation. Forced Oscillations [509] --
4. Additional Remarks on Differential Equations [519] --
Summary of Important Theorems and Formulae [529] --
Miscellaneous Examples [549] --
Answers and Hints [571] --
Index [611] --
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