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

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

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

### 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 Formulæ