# Book:Richard Courant/Differential and Integral Calculus/Volume II

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## Richard Courant:

## Richard Courant: *Differential and Integral Calculus, Volume $\text { II }$*

Published $\text {1936}$, **Blackie** (translated by J.E. McShane).

### Subject Matter

### Contents

- Preface

- Chapter I Preliminary Remarks on Analytical Geometry and Vector Analysis
- 1. Rectangular Co-ordinates and Vectors
- 2. The Area of a Triangle, the Volume of a Tetrahedron, the Vector Multiplication of Vectors
- 3. Simple Theorems on Determinants of the Second and Third Order
- 4. Affine Transformations and the Multiplication of Determinants

- Chapter II Functions of Several Variables and their Derivatives
- 1. The Concept of Function in the Case of Several Variables
- 2. Continuity
- 3. The Derivatives of a Function
- 4. The Total Differential of a Function and its Geometrical Meaning
- 5. Functions of Functions (Compound Functions) and the Introduction of New Independent Variables
- 6. The Mean Value Theorem and Taylor's Theorem for Functions of Several Variables
- 7. The Application of Vector Methods

- Appendix
- 1. The Principle of the Point of Accumulation in Several Dimensions and its Applications
- 2. The Concept of Limit for Functions of Several Variables
- 3. Homogeneous Functions

- Chapter III Developments and Applications of the Differential Calculus
- 1. Implicit Functions
- 2. Curves and Surfaces in Implicit Form
- 3. Systems of Functions, Transformations, and Mappings
- 4. Applications
- 5. Families of Curves, Families of Surfaces, and their Envelopes
- 6. Maxima and Minima

- Appendix
- 1. Sufficient Conditions for Extreme Value
- 2. Singular Points of Plane Curve
- 3. Singular Points of Surfaces
- 4. Connexion between Euler's and Lagrange's Representations of the Motion of a Fluid
- 5. Tangential Representation of a Closed Curve

- Chapter IV Multiple Integrals
- 1. Ordinary Integrals as Functions of a Parameter
- 2. The Integral of a Continuous Function over a Region of the Plane or of Space
- 3. Reduction of the Multiple Integral to Repeated Single Integrals
- 4. Transformation of Multiple Integrals
- 5. Improper Integrals
- 6. Geometrical Applications
- 7. Physical Applications

- Appendix
- 1. The Existence of the Multiple Integral
- 2. General Formula for the Area (or Volume) of a Region bounded by Segments of Straight Lines or Plane Areas (Guldin's Formula). The Polar Planimeter
- 3. Volumes and Areas in Space of any Number of Dimensions
- 4. Improper Integrals as Functions of a Parameter
- 5. The Fourier Integral
- 6. The Eulerian Integral (Gamma Function)
- 7. Differentiation and Integration to Fractional Order. Abel's Integral Equation
- 8. Note on the Definition of the Area of a Curved Surface

- Chapter V Integration over Regions in Several Dimensions
- 1. Line Integrals
- 2. Connexion between Line Integrals and Double Integrals in the Plane. (The Integral Theorems of Gauss, Stokes, and Green)
- 3. Interpretation and Applications of the Integral Theorems for the Plane
- 4. Surface Integrals
- 5. Gauss's Theorem and Green's Theorem in Space
- 6. Stokes's Theorem in Space
- 7. The Connexion between Differentiation and Integration for Several Variables

- Appendix
- 1. Remarks on Gauss's Theorem and Stokes's Theorem
- 2. Representation of a Source-free Vector Field as a Curl

- Chapter VI Differential Equations
- 1. The Differential Equation of the Motion of a Particle in Three Dimensions
- 2. Examples on the Mechanics of a Particle
- 3. Further Examples of Differential Equations
- 4. Linear Differential Equations
- 5. General Remarks on Differential Equations
- 6. The Potential of Attracting Charges
- 7. Further Examples of Partial Differential Equations

- Chapter VII Calculus of Variations
- 1. Introduction
- 2. Euler's Differential Equation in the Simplest Case
- 3. Generalizations

- Chapter VIII Functions of a Complex Variable
- 1. Introduction
- 2. Foundations of the Theory of Functions of a Complex Variable
- 3. The Integration of Analytic Functions
- 4. Cauchy's Formula and its Applications
- 5. Applications to Complex Integration (Contour Integration)
- 6. Many-valued Functions and Analytic Extension

- Supplement
- Real Numbers and the Concept of Limit
- Miscellaneous Examples
- Summary of Important Theorems and FormulĂ¦
- Answers and Hints

**Index**

## Source work progress

- 1936: Richard Courant:
*Differential and Integral Calculus: Volume $\text { II }$*... (previous) ... (next): Chapter $\text I$: Preliminary Remarks on Analytical Geometry and Vector Analysis: $1$. Rectangular Co-ordinates and Vectors: $2$. Directions and Vectors. FormulĂ¦ for Transforming Axes