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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.