Tensor Algebra and Analysis for Engineers With Applications to Differential Geometry of Curves and Surfaces 1st Edition by Paolo Vannucci 9811264821978 9811264825

1. Points and Vectors

1.1 Points and vectors

1.2 Scalar product, distance, orthogonality

1.3 Basis of V, expression of the scalar product

1.4 Applied vectors

1.5 Exercises

2. Second-Rank Tensors

2.1 Second-rank tensors

2.2 Dyads, tensor components

2.3 Tensor product

2.4 Transpose, symmetric and skew tensors

2.5 Trace, scalar product of tensors

2.6 Spherical and deviatoric parts

2.7 Determinant, inverse of a tensor

2.8 Eigenvalues and eigenvectors of a tensor

2.9 Skew tensors and cross product

2.10 Orientation of a basis

2.11 Rotations

2.12 Reflexions

2.13 Polar decomposition

2.14 Exercises

3. Fourth-Rank Tensors

3.1 Fourth-rank tensors

3.2 Dyads, tensor components

3.3 Conjugation product, transpose, symmetries

3.4 Trace and scalar product of fourth-rank tensors

3.5 Projectors and identities

3.6 Orthogonal conjugator

3.7 Rotations and symmetries

3.8 The Kelvin formalism

3.9 The polar formalism for plane tensors

3.10 Exercises

4. Tensor Analysis: Curves

4.1 Curves of points, vectors and tensors

4.2 Differentiation of curves

4.3 Integral of a curve of vectors and length of a curve

4.4 The Frenet–Serret basis

4.5 Curvature of a curve

4.6 The Frenet–Serret formula

4.7 The torsion of a curve

4.8 Osculating sphere and circle

4.9 Evolute, involute and envelopes of plane curves

4.10 The theorem of Bonnet

4.11 Canonic equations of a curve

4.12 Exercises

5. Tensor Analysis: Fields

5.1 Scalar, vector and tensor fields

5.2 Differentiation of fields, differential operators

5.3 Properties of the differential operators

5.4 Theorems on fields

5.5 Differential operators in Cartesian coordinates

5.6 Differential operators in cylindrical coordinates

5.7 Differential operators in spherical coordinates

5.8 Exercises

6. Curvilinear Coordinates

6.1 Introduction

6.2 Curvilinear coordinates, metric tensor

6.3 Co- and contravariant components

6.4 Spatial derivatives of fields in curvilinear coordinates

6.5 Exercises

7. Surfaces in E

7.1 Surfaces in E, coordinate lines and tangent planes

7.2 Surfaces of revolution

7.3 Ruled surfaces

7.4 First fundamental form of a surface

7.5 Second fundamental form of a surface

7.6 Curvatures of a surface

7.7 The theorem of Rodrigues

7.8 Classification of the points of a surface

7.9 Developable surfaces

7.10 Points of a surface of revolution

7.11 Lines of curvature, conjugated directions, asymptotic directions

7.12 Dupin’s conical curves

7.13 The Gauss–Weingarten equations

7.14 The theorema egregium

7.15 Minimal surfaces

7.16 Geodesics

7.17 The Gauss–Codazzi compatibility conditions

7.18 Exercises

Suggested Texts

Solutions to the Exercises

Index