Author | Courant, Richard. author |
---|---|
Title | Introduction to Calculus and Analysis [electronic resource] : Volume II / by Richard Courant, Fritz John |
Imprint | New York, NY : Springer New York, 1989 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-8958-3 |
Descript | XXV, 954 p. online resource |
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 Taylorโs 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 Eulerโs and Lagrangeโs Representation of the motion of a Fluid -- A.6 Tangential Representation of a Closed Curve and the Isoperimetric Inequality -- 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. Stokesโs Theorem -- 5.3 Formula for Integration by Parts in Two Dimensions. Greenโs 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 Gaussโs and Greenโs Theorems in Space -- 5.10 Stokesโs 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 Stokesโs Theorem -- A.4 Surfaces and Surface Integrals in Euclidean Spaces of Higher Dimensions -- A.5 Integrals over Simple Surfaces, Gaussโs 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 Laplaceโs 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 Cauchyโs Formula and Its Applications -- 8.5 Applications to Complex Integration (Contour Integration) -- 8.6 Many-Valued Functions and Analytic Extension -- List of Biographical Dates