Differential Forms

Hailed by Mathematical Review as "a superb text for an undergraduate course in advanced calculus" and "an excellent foundation for global and nonlinear algebra," this volume by a world-famous mathematician was originally published in France, where the science journal Annales declared it "precise and detailed, the style lucid and almost conversational . . . an outstanding text and work of reference."

Differential forms

Multilinear alternating mappings

Definition of an alternating multilinear mapping

9

(1)

Permutation groups

9

(1)

Properties of multilinear alternating mappings

10

(2)

Multiplication of multilinear alternating mappings

12

(2)

Properties of exterior multiplication

14

(2)

The exterior product of n linear forms

16

(1)

The case where E is of finite dimension

17

(1)

Differential forms

Definition of a differential form

18

(1)

Operations on differential forms

18

(2)

Exterior differentiation

20

(1)

Properties of exterior differentiation

21

(2)

Fundamental property of exterior differentiation

23

(1)

Differential forms on a finite dimensional space

24

(2)

Calculus of operations on canonically represented differential forms

26

(2)

Change of variable

28

(1)

Properties of the change of variable mapping φ*

29

(1)

Calculation of φ* in canonical form

30

(2)

Transitivity of the change of variable

32

(1)

Condition for a differential form to equal dα

33

(2)

Proof of Poincare's theorem

35

(5)

Curvilinear integral of a differential form of degree one

Paths of class C1

40

(1)

The curvilinear integral

40

(2)

Change of parameter

42

(1)

The case where ω is the differential of a function

43

(4)

The closed differential form of degree one

47

(1)

Primitive of a closed form along a path

48

(2)

Homotopy of two paths

50

(3)

Simply-connected domains

53

(1)

Integration of differential forms of degree > 1

Differentiable partitions of unity

54

(4)

Compact sets with boundary in the plane R2

58

(2)

Integral of a differential 2-form on a compact set with boundary

60

(2)

Stokes's theorem in the plane

62

(1)

Proof of Theorem 4.4.1 (Stokes's theorem)

63

(4)

Change of variable in double integrals

67

(4)

Varieties in Rn

71

(4)

Orientation of a variety

75

(1)

Integration of a differential 2-form over a compact, oriented variety of dimension 2 and of class C1

76

(3)

Multiple integrals

79

(2)

Differential forms on a variety M c Rn

81

(1)

The p-dimensional volume element of a variety M of dimension p (M c Rn)

82

(3)

Maxima and minima of a function on a variety

First order conditions

85

(1)

Second order conditions

86

(1)

Theorem of Frobenius

Introduction to the problem

87

(2)

The first existence theorem

89

(1)

The second existence theorem

90

(1)

Completion of the proof of the second existence theorem (Theorem 6.3.1)

91

(2)

The fundamental theorem

93

(1)

Interpretation in terms of differential forms

94

(4)

Exercises

98

(7)

Elements of the calculus of variations

Introduction to the problem

The space of curves of class C1

105

(1)

Functional associated with a curve

106

(2)

Example

108

(1)

A problem of minimum

108

(2)

Transformation of the condition for an extremum

110

(3)

Calculation of f' (g) u for an extremal

113

(1)

Study of the Euler equation. Existence of extremals. Examples

The Euler equation for the case E = Rn

114

(2)

Examples

116

(2)

The Lagrange equations in mechanics

118

(1)

More about the general case: the case where F(t, x, y) is independent of t

119

(1)

The case where F(x, y) is a homogeneous quadratic function of y

120

(2)

Geodesics on a variety

122

(2)

Extremum problems for curves constrained to lie on a variety

124

(3)

Transformation of the preceding condition

127

(1)

Problems in two dimensions

Introduction of the problem

128

(2)

Transformation of the condition for an extremum

130

(2)

Exercises

132

(6)

Applications of the moving frame method to the theory of curves and surfaces

The moving frame

Definition of the differential forms ω1, and ωij

138

(1)

Relations satisfied by the forms ωi and ωij

139

(1)

Orthonormal frames

140

(1)

The Frenet frame of an oriented curve in R3

141

(1)

The Darboux frame of an oriented curve C traced on an oriented surface S in R3

142

(2)

Calculation of the geodesic curvature, normal curvature and geodesic torsion

144

(2)

3-Parameter family of frames associated with a surface in R3

The variety of frames of an oriented surface

146

(1)

The equations of motion of a frame associated with an oriented surface

147

(1)

The element of area of the surface

148

(1)

The second fundamental quadratic form of the surface S

149

(1)

Calculation of the normal curvature and geodesic torsion in a given direction

150

(1)

Principal directions; lines of curvature

151

(2)

The differential form of geodesic curvature

153

(1)

Use of a field of frames

154

(1)

Parallel transport along a curve

155

(1)

Relation between total curvature and parallel transport

156

(2)

Calculation of the total curvature of a surface by means of the first fundamental form

158

(2)

Exercises

160

(4)

Index

164

(3)

Bibliography

167

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Differential Forms

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