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...
Main Authors: | , |
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Format: | eBook |
Language: | English |
Published: |
New York, NY
Springer New York
1989, 1989
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Edition: | 1st ed. 1989 |
Subjects: | |
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