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Introduction to Calculus and Analysis Volume II

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 su...

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Bibliographic Details
Main Authors: Courant, Richard, John, Fritz (Author)
Format: eBook
Language:English
Published: New York, NY Springer New York 1989, 1989
Edition:1st ed. 1989
Subjects:
Mathematical Analysis
Analysis
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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
  • 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
  • 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 Multiple Integrals
  • 4.1 Areas in the Plane
  • 4.2 Double Integrals
  • 4.3 Integrals over Regions in three and more Dimensions
  • 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

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