# Book:Richard Courant/Differential and Integral Calculus/Volume I/Second Edition

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## Richard Courant:

## Richard Courant: *Differential and Integral Calculus, Volume $\text { I }$ (2nd Edition)*

Published $\text {1934}$, **Blackie** (translated by E.J. McShane).

### Subject Matter

### Contents

- Preface to the First German Edition

- Preface to the English Edition (R. Courant), Cambridge, England,
*June, 1934.*)

- Preface to the Second English Edition (R. Courant), New Rochelle, N.Y.,,
*June*, 1937.)

- Introductory Remarks

- Chapter $\text {I}$ 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 $\text {I}$
- 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 $\text {II}$
- 1. Polar Co-ordinates
- 2. Remarks on Complex Numbers

- Chapter $\text {II}$ 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 Theorem 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 $\text {III}$ Differentiation and Integration of the Elementary Functions
- 1. The Simplest Rules for Differentiation and their Applications
- 2. The Corresponding Integral Formulæ
- 3. The Inverse Function and its Derivative
- 4. Differentiation of a Function of a Function
- 5. Maxima and Minima
- 6. The Logarithm and the Exponentiation Function
- 7. Some Applications of the Exponentiation 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 Formulæ

- Chapter $\text {IV}$ 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 $\text {V}$ 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 $\text {VI}$ 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 a Binomial Series Converges
- 4. Zeros and Infinities of Functions, and So-called Indeterminate Expressions

- Chapter $\text {VII}$ 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 $\text {VIII}$ 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 $\text {IX}$ 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 $\text {X}$ 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 $\text {XI}$ 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 Formulæ
- Miscellaneous Examples
- Answers and Hints
- Index

## Further Editions

## Source work progress

- 1937: Richard Courant:
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