Differential and Integral Calculus Vol. I

Description

PREFACE CONTENTS INTRODUCTORY REMARKS CHAPTER I INTRODUCTION 1. The Continuum of Numbers 2. The Concept of Function 3. More Detailed Study of the Elementary Functions 4. Functions of an Integral Variable. Sequences of Numbers 5. The Concept of the Limit of a Sequence 6. Further Discussion of the Concept of Limit 7. The Concept of Limit where the Variable is Continuous 8. The Concept of Continuity APPENDIX I Preliminary Remarks 1. The Principle of the Point of Accumulation and its Applications 2. Theorems on Continuous Functions 3. Some Remarks on the Elementary Functions APPENDIX II 1. Polar Co-ordinates 2. Remarks on Complex Numbers CHAPTER II THE FUNDAMENTAL IDEAS OF THE INTEGRAL AND DIFFERENTIAL CALCULUS 1. The Definite Integral 2. Examples 3. The Derivative 4. The Indefinite Integral, the Primitive Function, and the Fundamental Theorems of the Differential and Integral Calculus 5. Simple Methods of Graphical Integration 6. Further Remarks on the Connexion between the Integral and the Derivative 7. The Estimation of Integrals and the Mean Value Theorem of the Integral Calculus APPENDIX 1. The Existence of the Definite Integral of a Continuous Function 2. The Relation between the Mean Value Theorem of the Differential Calculus and the Mean Value Theorem of the Integral Calculus CHAPTER III DIFFERENTIATION AND INTEGRATION OF THE ELEMENTARY FUNCTIONS 1. The Simplest Rules for Differentiation and their Applications 2. The Corresponding Integral Formulae 3. The Inverse Function and its Derivative 4. Differentiation of a Function of a Function 5. Maxima and Minima 6. The Logarithm and the Exponential Function 7. Some Applications of the Exponential Function 8. The Hyperbolic Functions 9. The Order of Magnitude of Functions APPENDIX 1. Some Special Functions 2. Remarks on the Differentiability of Functions 3. Some Special Formulae CHAPTER IV FURTHER DEVELOPMENT OF THE INTEGRAL CALCULUS 1. Elementary Integrals 2. The Method of Substitution 3. Further Examples of the Substitution Method 4. Integration by Parts 5. Integration of Rational Functions 6. Integration of Some Other Classes of Functions 7. Remarks on Functions which are not Integrable in Terms of Elementary Functions 8. Extension of the Concept of Integral. Improper Integrals APPENDIX The Second Mean Value Theorem of the Integral Calculus CHAPTER V APPLICATIONS 1. Representation of Curves 2. Applications to the Theory of Plane Curves 3. Examples 4. Some very Simple Problems in the Mechanics of a Particle 5. Further Applications: Particle sliding down a Curve 6. Work APPENDIX 1. Properties of the Evolute 2. Areas bounded by Closed Curves CHAPTER VI TAYLOR'S THEOREM AND THE APPROXIMATE EXPRESSION OF FUNCTIONS BY POLYNOMIALS 1. The Logarithm and the Inverse Tangent 2. Taylor's Theorem 3. Applications. Expansions of the Elementary Functions 4. Geometrical Applications APPENDIX 1. Example of a Function which cannot be expanded in a Taylor Series 2. Proof that e is Irrational 3. Proof that the Binomial Series Converges 4. Zeros and Infinities of Functions, and So-called Indeterminate Expressions CHAPTER VII NUMERICAL METHODS Preliminary Remarks 1. Numerical Integration 2. Applications of the Mean Value Theorem and of Taylor's Theorem. The Calculus of Errors 3. Numerical Solution of Equations APPENDIX Stirling's Formula CHAPTER VIII INFINITE SERIES AND OTHER LIMITING PROCESSES Preliminary Remarks 1. The Concepts of Convergence and Divergence 2. Tests for Convergence and Divergence 3. Sequences and Series of Functions 4. Uniform and Non-uniform Convergence 5. Power Series 6. Expansion of Given Functions in Power Series. Method of Undetermined Coefficients. Examples 7. Power Series with Complex Terms APPENDIX 1. Multiplication and Division of Series 2. Infinite Series and Improper Integrals 3. Infinite Products 4. Series involving Bernoulli's Numbers CHAPTER IX FOURIER SERIES 1. Periodic Functions 2. Use of Complex Notation 3. Fourier Series 4. Examples of Fourier Series 5. The Convergence of Fourier Series APPENDIX Integration of Fourier Series CHAPTER X A SKETCH OF THE THEORY OF FUNCTIONS OF SEVERAL VARIABLES 1. The Concept of Function in the Case of Several Variables 2. Continuity 3. The Derivatives of a Function of Several Variables 4. The Chain Rule and the Differentiation of Inverse Functions 5. Implicit Functions 6. Multiple and Repeated Integrals CHAPTER XI THE DIFFERENTIAL EQUATIONS FOR THE SIMPLEST TYPES OF VIBRATION 1. Vibration Problems of Mechanics and Physics 2. Solution of the Homogeneous Equation. Free Oscillations 3. The Non-homogeneous Equation. Forced Oscillations 4. Additional Remarks on Differential Equations SUMMARY OF IMPORTANT THEOREMS AND FORMULAE MISCELLANEOUS EXAMPLES ANSWERS AND HINTS INDEX