In modern theoretical and applied mechanics, tensors and differential geometry are two almost essential tools. Unfortunately, in university courses for engineering and mechanics students, these topics are often poorly treated or even completely ignored. At the same time, many existing, very complete texts on tensors or differential geometry are so advanced and written in abstract language that discourage young readers looking for an introduction to these topics specifically oriented to engineering applications.This textbook, mainly addressed to graduate students and young researchers in mechanics, is an attempt to fill the gap. Its aim is to introduce the reader to the modern mathematical tools and language of tensors, with special applications to the differential geometry of curves and surfaces in the Euclidean space. The exposition of the matter is sober, directly oriented to problems that are ordinarily found in mechanics and engineering. Also, the language and symbols are tailored to those usually employed in modern texts of continuum mechanics.Though not exhaustive, as any primer textbook, this volume constitutes a coherent, self-contained introduction to the mathematical tools and results necessary in modern continuum mechanics, concerning vectors, 2nd- and 4th-rank tensors, curves, fields, curvilinear coordinates, and surfaces in the Euclidean space. More than 100 exercises are proposed to the reader, many of them complete the theoretical part through additional results and proofs. To accompany the reader in learning, all the exercises are entirely developed and solved at the end of the book.
Author(s): Paolo Vannucci
Series: Contemporary Mathematics and Its Applications: Monographs, Expositions and Lecture Notes, 5
Publisher: World Scientific
Year: 2023
Language: English
Pages: 229
City: Singapore
Contents
Preface
About the Author
Acknowledgments
List of Symbols
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
