The new Chapter 1 contains all the fundamental properties of linear differential forms and their integrals. These prepare the reader for the introduction to higher-order exterior differential forms added to Chapter 3. Also found now in Chapter 3 are a new proof of the implicit function theorem by successive approximations and a discus­ sion of numbers of critical points and of indices of vector fields in two dimensions. Extensive additions were made to the fundamental properties of multiple integrals in Chapters 4 and 5. Here one is faced with a familiar difficulty: integrals over a manifold M, defined easily enough by subdividing M into convenient pieces, must be shown to be inde­ pendent of the particular subdivision. This is resolved by the sys­ tematic use of the family of Jordan measurable sets with its finite intersection property and of partitions of unity. In order to minimize topological complications, only manifolds imbedded smoothly into Euclidean space are considered. The notion of "orientation" of a manifold is studied in the detail needed for the discussion of integrals of exterior differential forms and of their additivity properties. On this basis, proofs are given for the divergence theorem and for Stokes's theorem in n dimensions. To the section on Fourier integrals in Chapter 4 there has been added a discussion of Parseval's identity and of multiple Fourier integrals.

Table of Contents:
1 Functions of Several Variables and Their Derivatives.- 1.1 Points and Points Sets in the Plane and in Space.- 1.2 Functions of Several Independent Variables.- 1.3 Continuity.- 1.4 The Partial Derivatives of a Function.- 1.5 The Differential of a Function and Its Geometrical Meaning.- 1.6 Functions of Functions (Compound Functions) and the Introduction of New Independent Variables.- 1.7 The Mean Value Theorem and Taylor’s Theorem for Functions of Several Variables.- 1.8 Integrals of a Function Depending on a Parameter.- 1.9 Differentials and Line Integrals.- 1.10 The Fundamental Theorem on Integrability of Linear Differential Forms.- A.1. The Principle of the Point of Accumulation in Several Dimensions and Its Applications.- A.2. Basic Properties of Continuous Functions.- A.3. Basic Notions of the Theory of Point Sets.- A.4. Homogeneous functions.- 2 Vectors, Matrices, Linear Transformations.- 2.1 Operations with Vectors.- 2.2 Matrices and Linear Transformations.- 2.3 Determinants.- 2.4 Geometrical Interpretation of Determinants.- 2.5 Vector Notions in Analysis.- 3 Developments and Applications of the Differential Calculus.- 3.1 Implicit Functions.- 3.2 Curves and Surfaces in Implicit Form.- 3.3 Systems of Functions, Transformations, and Mappings.- 3.4 Applications.- 3.5 Families of Curves, Families of Surfaces, and Their Envelopes.- 3.6 Alternating Differential Forms.- 3.7 Maxima and Minima.- A.1 Sufficient Conditions for Extreme Values.- A.2 Numbers of Critical Points Related to Indices of a Vector Field.- A.3 Singular Points of Plane Curves.- A.4 Singular Points of Surfaces.- A.5 Connection Between Euler’s and Lagrange’s Representation of the motion of a Fluid.- A.6 Tangential Representation of a Closed Curve and the Isoperimetric Inequality.- 4 MultipleIntegrals.- 4.1 Areas in the Plane.- 4.2 Double Integrals.- 4.3 Integrals over Regions in three and more Dimensions.- 4.4 Space Differentiation. Mass and Density.- 4.5 Reduction of the Multiple Integral to Repeated Single Integrals.- 4.6 Transformation of Multiple Integrals.- 4.7 Improper Multiple Integrals.- 4.8 Geometrical Applications.- 4.9 Physical Applications.- 4.10 Multiple Integrals in Curvilinear Coordinates.- 4.11 Volumes and Surface Areas in Any Number of Dimensions.- 4.12 Improper Single Integrals as Functions of a Parameter.- 4.13 The Fourier Integral.- 4.14 The Eulerian Integrals (Gamma Function).- Appendix: Detailed Analysis of the Process of Integration.- A.1 Area.- A.2 Integrals of Functions of Several Variables.- A.3 Transformation of Areas and Integrals.- A.4 Note on the Definition of the Area of a Curved Surface.- 5 Relations Between Surface and Volume Integrals.- 5.1 Connection Between Line Integrals and Double Integrals in the Plane (The Integral Theorems of Gauss, Stokes, and Green).- 5.2 Vector Form of the Divergence Theorem. Stokes’s Theorem.- 5.3 Formula for Integration by Parts in Two Dimensions. Green’s Theorem.- 5.4 The Divergence Theorem Applied to the Transformation of Double Integrals.- 5.5 Area Differentiation. Transformation of ?u to Polar Coordinates.- 5.6 Interpretation of the Formulae of Gauss and Stokes by Two-Dimensional Flows.- 5.7 Orientation of Surfaces.- 5.8 Integrals of Differential Forms and of Scalars over Surfaces.- 5.9 Gauss’s and Green’s Theorems in Space.- 5.10 Stokes’s Theorem in Space.- 5.11 Integral Identities in Higher Dimensions.- Appendix: General Theory of Surfaces and of Surface Integals.- A.1 Surfaces and Surface Integrals in Three dimensions.- A.2 The Divergence Theorem.- A.3 Stokes’s Theorem.-A.4 Surfaces and Surface Integrals in Euclidean Spaces of Higher Dimensions.- A.5 Integrals over Simple Surfaces, Gauss’s Divergence Theorem, and the General Stokes Formula in Higher Dimensions.- 6 Differential Equations.- 6.1 The Differential Equations for the Motion of a Particle in Three Dimensions.- 6.2 The General Linear Differential Equation of the First Order.- 6.3 Linear Differential Equations of Higher Order.- 6.4 General Differential Equations of the First Order.- 6.5 Systems of Differential Equations and Differential Equations of Higher Order.- 6.6 Integration by the Method of Undermined Coefficients.- 6.7 The Potential of Attracting Charges and Laplace’s Equation.- 6.8 Further Examples of Partial Differential Equations from Mathematical Physics.- 7 Calculus of Variations.- 7.1 Functions and Their Extrema.- 7.2 Necessary conditions for Extreme Values of a Functional.- 7.3 Generalizations.- 7.4 Problems Involving Subsidiary Conditions. Lagrange Multipliers.- 8 Functions of a Complex Variable.- 8.1 Complex Functions Represented by Power Series.- 8.2 Foundations of the General Theory of Functions of a Complex Variable.- 8.3 The Integration of Analytic Functions.- 8.4 Cauchy’s Formula and Its Applications.- 8.5 Applications to Complex Integration (Contour Integration).- 8.6 Many-Valued Functions and Analytic Extension.- List of Biographical Dates.