Contents
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1 The Taylor Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1 Case of a Function of One Variable . . . . . . . . . . . . . .
1
1.1.2 Cases of Functions of Several Variables . . . . . . . . . . .
2
1.2 Integrals with Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3 Fundamentals of the Theory of Field . . . . . . . . . . . . . . . . . . . .
8
1.3.1 Directional Derivative and Gradient . . . . . . . . . . . . . . .
8
1.3.2 Flux and Divergence of Vector Field . . . . . . . . . . . . . . 15
1.3.3 The Gauss Theorem and Green’s Formulae . . . . . . . . . 20
1.3.4 Circulation and Rotation of Vector Field . . . . . . . . . . . 28
1.3.5 The Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.3.6 The United Gauss Formula Expressed by Gradient,
Divergence and Rotation . . . . . . . . . . . . . . . . . . . . . . . 39
1.4 Coordinate Transformations Between Rectangular Coordinate
System and Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.5 Fundamental Lemmas of the Calculus of Variations . . . . . . . . . 44
1.6 Summation Convention, Kronecker Delta and Permutation
Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.7 Basic Conceptions of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.7.1 Rotation Transformations of Rectangle Coordinates . . . 58
1.7.2 The Cartesian Second Order Tensors . . . . . . . . . . . . . . 60
1.7.3 Algebraic Operations of Cartesian Tensors . . . . . . . . . . 62
1.7.4 Quotient Laws of Tensors . . . . . . . . . . . . . . . . . . . . . . 64
1.7.5 Principal Axes, Characteristic Values and Invariants
of Second Order Tensors . . . . . . . . . . . . . . . . . . . . . . 65
1.7.6 Differential Operations of the Cartesian Tensors . . . . . . 67
1.8 Some Inequalities in Common Use . . . . . . . . . . . . . . . . . . . . . 69
1.9 Introduction to the Famous Scientists . . . . . . . . . . . . . . . . . . . . 75
xiii
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1 The Taylor Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1 Case of a Function of One Variable . . . . . . . . . . . . . .
1
1.1.2 Cases of Functions of Several Variables . . . . . . . . . . .
2
1.2 Integrals with Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3 Fundamentals of the Theory of Field . . . . . . . . . . . . . . . . . . . .
8
1.3.1 Directional Derivative and Gradient . . . . . . . . . . . . . . .
8
1.3.2 Flux and Divergence of Vector Field . . . . . . . . . . . . . . 15
1.3.3 The Gauss Theorem and Green’s Formulae . . . . . . . . . 20
1.3.4 Circulation and Rotation of Vector Field . . . . . . . . . . . 28
1.3.5 The Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.3.6 The United Gauss Formula Expressed by Gradient,
Divergence and Rotation . . . . . . . . . . . . . . . . . . . . . . . 39
1.4 Coordinate Transformations Between Rectangular Coordinate
System and Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 40
1.5 Fundamental Lemmas of the Calculus of Variations . . . . . . . . . 44
1.6 Summation Convention, Kronecker Delta and Permutation
Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1.7 Basic Conceptions of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . 58
1.7.1 Rotation Transformations of Rectangle Coordinates . . . 58
1.7.2 The Cartesian Second Order Tensors . . . . . . . . . . . . . . 60
1.7.3 Algebraic Operations of Cartesian Tensors . . . . . . . . . . 62
1.7.4 Quotient Laws of Tensors . . . . . . . . . . . . . . . . . . . . . . 64
1.7.5 Principal Axes, Characteristic Values and Invariants
of Second Order Tensors . . . . . . . . . . . . . . . . . . . . . . 65
1.7.6 Differential Operations of the Cartesian Tensors . . . . . . 67
1.8 Some Inequalities in Common Use . . . . . . . . . . . . . . . . . . . . . 69
1.9 Introduction to the Famous Scientists . . . . . . . . . . . . . . . . . . . . 75
xiii