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 In-dependent 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.- Appendix 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.- Appendix 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 360 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 Isoperi-metricInequality.- 4 Multiple Integrals.- 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 Au 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.I Surfaces and Surface Integrals in Three dimensions.- A.2 The Divergence Theorem.- A.3Stokes’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.