Solutions of Exercises of Principles of Tensor Calculus

239 Pages·2018·2.44 MB·English

by  Taha Sochi

Preface
This book contains the solutions of all the exercises of my book: Principles of
Tensor Calculus. These solutions are sufficiently simplified and detailed for the
benefit of readers of all levels particularly those at introductory levels.
Taha Sochi
London, September 2018

Table of Contents
Preface
Nomenclature
Chapter 1 Preliminaries
Chapter 2 Spaces, Coordinate Systems and Transformations
Chapter 3 Tensors
Chapter 4 Special Tensors
Chapter 5 Tensor Differentiation
Chapter 6 Differential Operations
Chapter 7 Tensors in Application
Author Notes
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. ũi, Ã) transformed quantity
i
○ inner or outer product operator
⊥ perpendicular to
1D, 2D, 3D, nD one-, two-, three-, n-dimensional
δ ⁄ δt absolute derivative operator with respect to t
∂ and ∇ partial derivative operator with respect to ith variable
i i
∂ covariant derivative operator with respect to ith variable
;i
[ij, k] Christoffel symbol of 1st kind
A area
B, B Finger strain tensor
ij
B − 1, B  − 1 Cauchy strain tensor
ij
C curve
Cn of class n
d, d displacement vector
i
det determinant of matrix
diag[⋯] diagonal matrix with embraced diagonal elements
dr differential of position vector
ds length of infinitesimal element of curve
dσ area of infinitesimal element of surface
dτ volume of infinitesimal element of space
e ith 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 rθ φφ
e , e , e basis vectors of cylindrical coordinate system
ρ φ z
e , e , ⋯, e unit dyads of cylindrical coordinate system
ρρ ρφ zz
E, E first displacement gradient tensor
ij
E, Ei ith covariant and contravariant basis vectors
i
ℰ ith orthonormalized covariant basis vector
i
Eq./Eqs. Equation/Equations
g determinant of covariant metric tensor
g metric tensor
g , gij, gj covariant, contravariant and mixed metric tensor or its
ij i
components
g , g , ⋯g coefficients of covariant metric tensor
11 12 nn
g11, g12, ⋯gnn coefficients of contravariant metric tensor
h scale factor for ith 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
point
point
P
P(n, k) k-permutations of n objects
qi ith coordinate of orthogonal coordinate system
q ith unit basis vector of orthogonal coordinate system
i
r position vector
ℛ Ricci curvature scalar
R , Ri Ricci curvature tensor of 1st and 2nd kind
ij j
R , Ri Riemann-Christoffel curvature tensor of 1st and 2nd 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
ui ith coordinate of general coordinate system
v, v velocity vector
i
V volume
weight of relative tensor
w weight of relative tensor
x, xi ith Cartesian coordinate
i
x’, x ith Cartesian coordinate of particle at past and present times
i i
x, y, z coordinates of 3D space (mainly Cartesian)
γ, γ infinitesimal strain tensor
ij
γ̇ rate of strain tensor
Γk Christoffel symbol of 2nd kind
ij
δ Kronecker delta tensor
δ , δij, δj covariant, contravariant and mixed ordinary Kronecker
ij i
delta
δij , δijk , generalized Kronecker delta in 2D, 3D and nD space
kl lmn
δi1…in
j1…jn
Δ, Δ second displacement gradient tensor
ij
ϵ , ϵ , ϵ covariant relative permutation tensor in 2D, 3D and nD
ij ijk i1…in
space
ϵij, ϵijk, ϵi1…in contravariant relative permutation tensor in 2D, 3D and nD
space
ε , ε , ε covariant absolute permutation tensor in 2D, 3D and nD
ij ijk i1…in
space
εij, εijk,εi1…in contravariant absolute permutation tensor in 2D, 3D and
nD space
ρ, φ coordinates of plane polar coordinate system
coordinates of cylindrical coordinate system
coordinates of cylindrical coordinate system
ρ, φ, z
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
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: Γi or the generalized Kronecker delta as: δi1…in instead of
jk j1…jn
their normal look as: and .

4. Due to the difficulty of converting the ordinary integral symbol
(i.e. \int) correctly to the mobi format we use the following integral
symbol ⨏ (i.e. \fint) as substitute in the textual mathematical expressions. In
brief, all the integral symbols in this book represent ordinary
integrals.

5. Some symbols in the mobi version are not the
same as in the paper version. The reader therefore should consult the
Nomenclature of the given version for clarification.
Chapter 1
Preliminaries
1. Differentiate between the symbols used to label scalars, vectors and tensors
of rank  > 1.
Answer (see Footnote 1 in § 8↓):
Scalars: non-indexed lower case light face italic Latin letters (e.g. f and h)
are used to label scalars.
Vectors: non-indexed lower or upper case bold face non-italic Latin letters
(e.g. a and A) are used to label vectors in symbolic notation with the
exception of
the basis vectors where indexed bold face lower or upper case
non-italic symbols (e.g. e and Ei) are used.
1
Tensors of rank  > 1: non-indexed upper case bold face non-italic Latin
letters (e.g. A and B) are used to label tensors of rank  > 1 in symbolic
notation.
Indexed light face italic Latin symbols (e.g. a and Bjk) are used to denote
i i
tensors of rank  > 0 (i.e. vectors and tensors of rank  > 1) in their explicit
tensor form, i.e. index notation.
2. What the comma and semicolon in Ajk and A mean?
, i k;i
Answer: The comma means partial derivative with respect to the variable
whose index follows the comma (i.e. the ith variable in Ajk ), while
, i
semicolon means covariant derivative with respect to the variable whose
index follows the semicolon (i.e. the ith variable in A ).
k;i
3. State the summation convention and explain its conditions. To what type of
indices this convention applies?
Answer: According to the summation convention, dummy indices