Principles of Tensor Calculus: Tensor Calculus

354 Pages·2017·7.73 MB·English

by  Taha Sochi

Preface
This book is based on my previous book: Tensor Calculus Made Simple, where
the development of tensor calculus concepts and techniques are continued at a
higher level. In the present book, we continue the discussion of the main topics
of the subject at a more advanced level expanding, when necessary, some topics
and developing further concepts and techniques. The purpose of the present book
is to solidify, generalize, fill the gaps and make more rigorous what have been
presented in the previous book.
Unlike the previous book which is largely based on a Cartesian approach, the
formulation in the present book is largely based on assuming an underlying
general coordinate system although some example sections are still based on a
Cartesian approach for the sake of simplicity and clarity. The reader will be
notified about the underlying system in the given formulation. We also provide a
sample of formal proofs to familiarize the reader with the tensor techniques.
However, due to the preset objectives and the intended size of the book, we do
not offer comprehensive proofs and complete theoretical foundations for the
provided materials although we generally try to justify many of the given
formulations descriptively or by interlinking to related formulations or by similar
pedagogical techniques. This may be seen as a more friendly method for
constructing and establishing the abstract concepts and techniques of tensor
calculus.
The book is furnished with an index in the end of the book as well as rather
detailed sets of exercises in the end of each chapter to provide useful revision
and practice. To facilitate linking related concepts and sections, and hence
ensure better understanding of the given materials, cross referencing, which is
hyperlinked for the ebook users, is used extensively throughout the book. The
book also contains a number of graphic illustrations to help the readers to
visualize the ideas and understand the subtle concepts.
The book can be used as a text for an introductory or an intermediate level
course on tensor calculus. The familiarity with the materials presented in the
previous book will be an advantage although it is not necessary for someone
with a reasonable mathematical background. Moreover, the main materials of the
previous book are absorbed within the structure of the present book for the sake
of completeness and to make the book rather self-contained considering the
predetermined objectives. I hope I achieved these goals.
Taha Sochi
Taha Sochi
London, August 2017
Table of Contents
Preface
Nomenclature
1: Preliminaries
1.1: General Conventions and Notations
1.2: General Background about Tensors
1.3: Exercises and Revision
2: Spaces, Coordinate Systems and Transformations
2.1: Spaces
2.2: Coordinate Systems
2.2.1: Rectilinear and Curvilinear Coordinate Systems
2.2.2: Orthogonal Coordinate Systems
2.2.3: Homogeneous Coordinate Systems
2.3: Transformations
2.3.1: Proper and Improper Transformations
2.3.2: Active and Passive Transformations
2.3.3: Orthogonal Transformations
2.3.4: Linear and Nonlinear Transformations
2.4: Coordinate Curves and Coordinate Surfaces
2.5: Scale Factors
2.6: Basis Vectors and Their Relation to the Metric and Jacobian
2.7: Relationship between Space, Coordinates and Metric
2.8: Exercises and Revision
3: Tensors
3.1: Tensor Types
3.1.1: Covariant and Contravariant Tensors
3.1.2: True and Pseudo Tensors
3.1.3: Absolute and Relative Tensors
3.1.4: Isotropic and Anisotropic Tensors
3.1.5: Symmetric and Antisymmetric Tensors
3.1.6: General and Affine Tensors
3.2: Tensor Operations
3.2.1: Addition and Subtraction
3.2.2: Multiplication of Tensor by Scalar
3.2.3: Tensor Multiplication
3.2.4: Contraction
3.2.5: Inner Product
3.2.6: Permutation
3.2.7: Tensor Test and Quotient Rule
3.3: Tensor Representations
3.4: Exercises and Revision
4: Special Tensors
4.1: Kronecker delta Tensor
4.2: Permutation Tensor
4.3: Identities Involving Kronecker or/and Permutation Tensors
4.3.1: Identities Involving Kronecker delta Tensor
4.3.2: Identities Involving Permutation Tensor
4.3.3: Identities Involving Kronecker and Permutation Tensors
4.4: Generalized Kronecker delta Tensor
4.5: Metric Tensor
4.6: Definitions Involving Special Tensors
4.6.1: Dot Product
4.6.2: Magnitude of Vector
4.6.3: Angle between Vectors
4.6.4: Cross Product
4.6.5: Scalar Triple Product
4.6.6: Vector Triple Product
4.6.7: Determinant of Matrix
4.6.8: Length
4.6.9: Area
4.6.10: Volume
4.7: Exercises and Revision
5: Tensor Differentiation
5.1: Christoffel Symbols
5.2: Covariant Differentiation
5.3: Absolute Differentiation
5.4: Exercises and Revision
6: Differential Operations
6.1: Cartesian Coordinate System
6.1.1: Operators
6.1.2: Gradient
6.1.3: Divergence
6.1.4: Curl
6.1.5: Laplacian
6.2: General Coordinate System
6.2.1: Operators
6.2.2: Gradient
6.2.3: Divergence
6.2.4: Curl
6.2.5: Laplacian
6.3: Orthogonal Coordinate System
6.3.1: Operators
6.3.2: Gradient
6.3.3: Divergence
6.3.4: Curl
6.3.5: Laplacian
6.4: Cylindrical Coordinate System
6.4.1: Operators
6.4.2: Gradient
6.4.3: Divergence
6.4.4: Curl
6.4.5: Laplacian
6.5: Spherical Coordinate System
6.5.1: Operators
6.5.2: Gradient
6.5.3: Divergence
6.5.4: Curl
6.5.5: Laplacian
6.6: Exercises and Revision
7: Tensors in Application
7.1: Tensors in Mathematics
7.1.1: Common Definitions in Tensor Notation
7.1.2: Scalar Invariants of Tensors
7.1.3: Common Identities in Vector and Tensor Notation
7.1.4: Integral Theorems in Tensor Notation
7.1.5: Examples of Using Tensor Techniques to Prove Identities
7.2: Tensors in Geometry
7.2.1: Riemann-Christoffel Curvature Tensor
7.2.2: Bianchi Identities
7.2.3: Ricci Curvature Tensor and Scalar
7.3: Tensors in Science
7.3.1: Infinitesimal Strain Tensor
7.3.2: Stress Tensor
7.3.3: Displacement Gradient Tensors
7.3.4: Finger Strain Tensor
7.3.5: Cauchy Strain Tensor
7.3.6: Velocity Gradient Tensor
7.3.7: Rate of Strain Tensor
7.3.8: Vorticity Tensor
7.4: Exercises and Revision
References
Author Notes
8: Footnotes
Nomenclature
In the following list, we define the common symbols, notations and
abbreviations which are used in the book as a quick reference for the reader.
∇ nabla differential operator
∇ and ∇; covariant and contravariant differential operators
;
∇f gradient of scalar f
∇⋅A divergence of tensor A
∇ × A curl of tensor A
∇2 , ∂ , ∇ Laplacian operator
ii ii
∇v , ∂ v velocity gradient tensor
i j
, (subscript) partial derivative with respect to following index(es)
; (subscript) covariant derivative with respect to following index(es)
hat (e.g. Â , Ê ) physical representation or normalized vector
i i
bar (e.g. ũ i , Ã ) transformed quantity
i
○ inner or outer product operator
⊥ perpendicular to
1D, 2D, 3D, n D one-, two-, three-, n -dimensional
δ  ⁄ δ t absolute derivative operator with respect to t
∂ and ∇ partial derivative operator with respect to i th variable
i i
∂ covariant derivative operator with respect to i th variable
;i
[ij , k ] Christoffel symbol of 1 st kind
A area
B , B Finger strain tensor
ij
B  − 1 , B  − 1 Cauchy strain tensor
ij
C curve
C n of class n
d , d displacement vector
i
det determinant of matrix
d r differential of position vector
ds length of infinitesimal element of curve
d σ area of infinitesimal element of surface
d τ volume of infinitesimal element of space
e i th vector of orthonormal vector set (usually Cartesian basis
i
set)
e , e , e basis vectors of spherical coordinate system
r θ φ
e , e , ⋯, e unit dyads of spherical coordinate system
rr r θ φ
φ
e , e , e basis vectors of cylindrical coordinate system
ρ φ z
e , e , ⋯, e unit dyads of cylindrical coordinate system
ρ ρ ρ φ
zz
E , E first displacement gradient tensor
ij
E , E i i th covariant and contravariant basis vectors
i
ℰ i th orthonormalized covariant basis vector
i
Eq./Eqs. Equation/Equations
g determinant of covariant metric tensor
g metric tensor
g , g ij , g j covariant, contravariant and mixed metric tensor or its
ij i
components
g , g , g coefficients of covariant metric tensor
11 12 22
g 11 , g 12 , g 22 coefficients of contravariant metric tensor
h scale factor for i th coordinate
i
iff if and only if
J Jacobian of transformation between two coordinate systems
J Jacobian matrix of transformation between two coordinate
systems
J  − 1 inverse Jacobian matrix of transformation
L length of curve
n , n normal vector to surface
i
P point
P (n , k ) k -permutations of n objects
q i i th coordinate of orthogonal coordinate system
q i th unit basis vector of orthogonal coordinate system
i
r position vector
ℛ Ricci curvature scalar
R , R i Ricci curvature tensor of 1 st and 2 nd kind
ij j
R , R i Riemann-Christoffel curvature tensor of 1 st and 2 nd kind
ijkl jkl
r , θ , φ coordinates of spherical coordinate system
S surface
S , S rate of strain tensor
ij
S̃ , S̃ vorticity tensor
ij
t time
T (superscript) transposition of matrix
T , T traction vector
i
tr trace of matrix
u i i th coordinate of general coordinate system
v , v velocity vector
i
V volume
w weight of relative tensor
x , x i i th Cartesian coordinate
i
x ’ , x i th Cartesian coordinate of particle at past and present times
i i
x , y , z coordinates of 3D space (mainly Cartesian)
infinitesimal strain tensor
γ , γ infinitesimal strain tensor
ij
γ ̇ rate of strain tensor
Γ k Christoffel symbol of 2 nd kind
ij
δ Kronecker delta tensor
δ , δ ij , δ j covariant, contravariant and mixed ordinary Kronecker delta
ij i
generalized Kronecker delta in n D space
Δ , Δ second displacement gradient tensor
ij
ϵ , ϵ , ϵ covariant relative permutation tensor in 2D, 3D, n D space
ij ijk i …i
1
n
ϵ ij , ϵ ijk , ϵ i 1 …i contravariant relative permutation tensor in 2D, 3D, n D
n space
ε , ε , ε covariant absolute permutation tensor in 2D, 3D, n D space
ij ijk i …i
1 n
ε ij , ε ijk , ε i 1 …i n contravariant absolute permutation tensor in 2D, 3D, n D
space
ρ , φ coordinates of plane polar coordinate system
ρ , φ , z coordinates of cylindrical coordinate system
σ , σ stress tensor
ij
ω vorticity tensor
Ω region of space
Note : due to the restrictions on the availability and visibility of symbols in the
mobi format, as well as similar formatting issues, we should draw the attention
of the ebook readers to the following points:
1. Bars over symbols, which are used in the printed version, were replaced by
tildes. However, for convenience we kept using the terms “barred” and
“unbarred” in the text to refer to the symbols with and without tildes.
2. The square root symbol in mobi is √ ( ) where the argument is contained
inside the parentheses. For example, the square root of g is symbolized as √ ( g )
.
3. In the mobi format, superscripts are automatically displayed before subscripts
unless certain measures are taken to force the opposite which may distort the
look of the symbol and may not even be the required format when the
superscripts and subscripts should be side by side which is not possible in the
mobi text and live equations. Therefore, for convenience and aesthetic reasons
we only forced the required order of the subscripts and superscripts or used
imaged symbols when it is necessary; otherwise we left the symbols to be
displayed according to the mobi choice although this may not be ideal like
displaying the Christoffel symbols of the second kind as: Γ i or the generalized
jk
Kronecker delta as: δ i 1 …i n instead of their normal look as: and
j …j
1 n
.