Taha Sochi - Principles of Tensor Calculus: Tensor Calculus

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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

London, August 2017

Table of Contents

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
, ii , ii	Laplacian operator
v , iv j	velocity gradient tensor
, (subscript)	partial derivative with respect to following index(es)
; (subscript)	covariant derivative with respect to following index(es)
hat (e.g. A i , E i )	physical representation or normalized vector
bar (e.g. u i , A i )	transformed quantity
inner or outer product operator
perpendicular to
1D, 2D, 3D, n D	one-, two-, three-, n -dimensional
t	absolute derivative operator with respect to t
i and i	partial derivative operator with respect to i th variable
;i	covariant derivative operator with respect to i th variable
[ij ,k ]	Christoffel symbol of 1st kind
A	area
B , B ij	Finger strain tensor
B 1 , B ij1	Cauchy strain tensor
C	curve
C n	of class n
d , d i	displacement vector
det	determinant of matrix
dr	differential of position vector
ds	length of infinitesimal element of curve
d	area of infinitesimal element of surface
d	volume of infinitesimal element of space
e i	i th vector of orthonormal vector set (usually Cartesian basis set)
e r ,e ,e	basis vectors of spherical coordinate system
e rr ,e r ,,e	unit dyads of spherical coordinate system
e ,e ,e z	basis vectors of cylindrical coordinate system
e ,e ,,e zz	unit dyads of cylindrical coordinate system
E , E ij	first displacement gradient tensor
E i , E i	i th covariant and contravariant basis vectors
i	i th orthonormalized covariant basis vector
Eq./Eqs.	Equation/Equations
g	determinant of covariant metric tensor
g	metric tensor
g ij , g ij , g ji	covariant, contravariant and mixed metric tensor or its components
g 11 ,g 12 ,g 22	coefficients of covariant metric tensor
g 11 ,g 12 ,g 22	coefficients of contravariant metric tensor
h i	scale factor for i th coordinate
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 i	normal vector to surface
P	point
P (n ,k )	k -permutations of n objects
q i	i th coordinate of orthogonal coordinate system
q i	i th unit basis vector of orthogonal coordinate system
r	position vector
Ricci curvature scalar
R ij , R ij	Ricci curvature tensor of 1st and 2nd kind
R ijkl , R ijkl	Riemann-Christoffel curvature tensor of 1st and 2nd kind
r , ,	coordinates of spherical coordinate system
S	surface
S , S ij	rate of strain tensor
S , S ij	vorticity tensor
t	time
T (superscript)	transposition of matrix
T , T i	traction vector
tr	trace of matrix
u i	i th coordinate of general coordinate system
v , v i	velocity vector
V	volume
w	weight of relative tensor

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