Mathematical Methods for Physics and Engineering

K. F. Riley, M. P. Hobson, S. J. Bence

Chapter 26

Tensors - all with Video Answers

Educators

Chapter Questions

Problem 1

Use the basic definition of a Cartesian tensor to show the following.
(a) That for any general, but fixed, $\phi$,
$$
\left(u_{1}, u_{2}\right)=\left(x_{1} \cos \phi-x_{2} \sin \phi, x_{1} \sin \phi+x_{2} \cos \phi\right)
$$
are the components of a first-order tensor in two dimensions.
(b) That
$$
\left(\begin{array}{cc}
x_{2}^{2} & x_{1} x_{2} \\
x_{1} x_{2} & x_{1}^{2}
\end{array}\right)
$$
is not a tensor of order 2 . To establish that a single element does not transform correctly is sufficient.

Jacob Fry

Jacob Fry

Numerade Educator

Problem 2

The components of two vectors, $\mathbf{A}$ and $\mathbf{B}$, and a second-order tensor, $\mathbf{T}$, are given in one coordinate system by
$$
A=\left(\begin{array}{l}
1 \\
0 \\
0
\end{array}\right), \quad B=\left(\begin{array}{l}
0 \\
1 \\
0
\end{array}\right), \quad \mathrm{T}=\left(\begin{array}{ccc}
2 & \sqrt{3} & 0 \\
\sqrt{3} & 4 & 0 \\
0 & 0 & 2
\end{array}\right)
$$
In a second coordinate system, obtained from the first by rotation, the components of $\mathbf{A}$ and $\mathbf{B}$ are
$$
\mathrm{A}^{\prime}=\frac{1}{2}\left(\begin{array}{l}
\sqrt{3} \\
0 \\
1
\end{array}\right), \quad \mathbf{B}^{\prime}=\frac{1}{2}\left(\begin{array}{c}
-1 \\
0 \\
\sqrt{3}
\end{array}\right)
$$
Find the components of $\mathbf{T}$ in this new coordinate system and hence evaluate, with a minimum of calculation,
$$
T_{i j} T_{j i}, \quad T_{k i} T_{j k} T_{i j}, \quad T_{i k} T_{m n} T_{n i} T_{k m^{*}}
$$

Jacob Fry

Numerade Educator

Problem 3

In section $26.3$ the transformation matrix for a rotation of the coordinate axes was derived, and this approach is used in the rest of the chapter. An alternative view is that of taking the coordinate axes as fixed and rotating the components of the system; this is equivalent to reversing the signs of all rotation angles.
Using this alternative view, determine the matrices representing (a) a positive rotation of $\pi / 4$ about the $x$-axis and (b) a rotation of $-\pi / 4$ about the $y$-axis. Determine the initial vector $\mathbf{r}$ which, when subjected to (a) followed by (b), finishes at $(3,2.1)$

Raj Bala

Numerade Educator

Problem 4

Show how to decompose the Cartesian tensor $T_{i j}$ into three tensors,
$$
T_{i j}=U_{i j}+V_{i j}+S_{i j}
$$
where $U_{i j}$ is symmetric and has zero trace, $V_{i j}$ is isotropic and $S_{i j}$ has only three independent components.

Jacob Fry

Numerade Educator

Problem 5

Use the quotient law discussed in section $26.7$ to show that the array
$$
\left(\begin{array}{ccc}
y^{2}+z^{2}-x^{2} & -2 x y & -2 x z \\
-2 y x & x^{2}+z^{2}-y^{2} & -2 y z \\
-2 z x & -2 z y & x^{2}+y^{2}-z^{2}
\end{array}\right)
$$
forms a second-order tensor.

Jacob Fry

Numerade Educator

Problem 6

Use tensor methods to establish the following vector identities:
(a) $\quad(\mathbf{u} \times \mathbf{v}) \times \mathbf{w}=(\mathbf{u} \cdot \mathbf{w}) \mathbf{v}-(\mathbf{v} \cdot \mathbf{w}) \mathbf{u}$
(b) $\operatorname{curl}(\phi \mathbf{u})=\phi \operatorname{curl} \mathbf{u}+(\operatorname{grad} \phi) \times \mathbf{u}$
(c) $\operatorname{div}(\mathbf{u} \times \mathbf{v})=\mathbf{v} \cdot \operatorname{curl} \mathbf{u}-\mathbf{u} \cdot \operatorname{curl} \mathbf{v}$
(d) $\operatorname{curl}(\mathbf{u} \times \mathbf{v})=(\mathbf{v} \cdot \mathbf{g r a d}) \mathbf{u}-(\mathbf{u} \cdot \mathbf{g r a d}) \mathbf{v}+\mathbf{u} \operatorname{div} \mathbf{v}-\mathbf{v} \operatorname{div} \mathbf{u} ;$
(e) $\operatorname{grad} \frac{1}{2}(\mathbf{u} \cdot \mathbf{u})=\mathbf{u} \times \operatorname{curl} \mathbf{u}+(\mathbf{u} \cdot \operatorname{grad}) \mathbf{u}$.

Jacob Fry

Numerade Educator

Problem 7

Use result (e) of the previous question and the general divergence theorem for tensors to show that, for a vector field $A$,
$$
\int_{S}\left[\mathbf{A}(\mathbf{A} \cdot d \mathbf{S})-\frac{1}{2} A^{2} d \mathbf{S}\right]=\int_{V}[\mathbf{A} \operatorname{div} \mathbf{A}-\mathbf{A} \times \operatorname{curl} \mathbf{A}] d V
$$
where $S$ is the surface enclosing volume $V$.

Jacob Fry

Numerade Educator

Problem 8

A column matrix a has components $a_{x}, a_{y}, a_{z}$ and $\mathrm{A}$ is the matrix with elements $A_{i j}=-\epsilon_{i j k} a_{k}$
(a) What is the relationship between column matrices $\mathrm{b}$ and $\mathrm{c}$ if $\mathrm{Ab}=\mathrm{c}$ ?
(b) Find the eigenvalues of $A$ and show that a is one of its eigenvectors. Explain why this must be so.

Jacob Fry

Numerade Educator

Problem 9

A column matrix a has components $a_{x}, a_{y}, a_{z}$ and $\mathrm{A}$ is the matrix with elements $A_{i j}=-\epsilon_{i j k} a_{k}$
(a) What is the relationship between column matrices $\mathrm{b}$ and $\mathrm{c}$ if $\mathrm{Ab}=\mathrm{c}$ ?
(b) Find the eigenvalues of $A$ and show that a is one of its eigenvectors. Explain why this must be so.

Jacob Fry

Numerade Educator

Problem 10

A symmetric second-order Cartesian tensor is defined by
$$
T_{i j}=\delta_{i j}-3 x_{i} x_{j}
$$
Evaluate the following surface integrals, each taken over the surface of the unit sphere:
(a) $\int T_{i j} d S$
(b) $\int T_{i k} T_{k j} d S$
(c) $\int x_{i} T_{j k} d S$.

Jacob Fry

Numerade Educator

Problem 11

Given a non-zero vector $\mathbf{v}$, find the value that should be assigned to $\alpha$ to make
$$
P_{i j}=\alpha v_{i} v_{j} \quad \text { and } \quad Q_{i j}=\delta_{i j}-\alpha v_{i} v_{j}
$$
into parallel and orthogonal projection tensors, respectively, i.e. tensors that satisfy, respectively, $P_{i j} v_{j}=v_{i}, P_{i j} u_{j}=0$ and $Q_{i j} v_{j}=0, Q_{i j} u_{j}=u_{i}$, for any vector u that is orthogonal to $\mathbf{v}$.

Show, in particular, that $Q_{i j}$ is unique, i.e. that if another tensor $T_{i j}$ has the same properties as $Q_{i j}$ then $\left(Q_{i j}-T_{i j}\right) w_{j}=0$ for any vector $\mathbf{w}$.

Jacob Fry

Numerade Educator

Problem 12

In four dimensions, define second-order antisymmetric tensors, $F_{i j}$ and $Q_{i j}$, and a first-order tensor, $S_{i}$, as follows:
(a) $F_{23}=H_{1}, Q_{23}=B_{1}$ and their cyclic permutations;
(b) $F_{i 4}=-D_{i}, Q_{i 4}=E_{i}$ for $i=1,2,3 ;$
(c) $S_{4}=\rho, S_{i}=J_{i}$ for $i=1,2,3$.
Then, taking $x_{4}$ as $t$ and the other symbols to have their usual meanings in electromagnetic theory, show that the equations $\sum_{j} \partial F_{i j} / \partial x_{j}=S_{i}$ and $\partial Q_{j k} / \partial x_{i}+$ $\partial Q_{k i} / \partial x_{j}+\partial Q_{i j} / \partial x_{k}=0$ reproduce Maxwell's equations. In the latter $i, j, k$ is any set of three subscripts selected from $1,2,3,4$, but chosen in such a way that they are all different.

Ahmad Reda

Numerade Educator

Problem 13

In a certain crystal the unit cell can be taken as six identical atoms lying at the corners of a regular octahedron. Convince yourself that these atoms can also be considered as lying at the centres of the faces of a cube and hence that the crystal has cubic symmetry. Use this result to prove that the conductivity tensor for the crystal, $\sigma_{i j}$, must be isotropic.

Benjamin Arndell

Benjamin Arndell

Numerade Educator

Problem 14

Assuming that the current density $\mathbf{j}$ and the electric field $\mathbf{E}$ appearing in equation (26.44) are first-order Cartesian tensors, show explicitly that the electrical conductivity tensor $\sigma_{i j}$ transforms according to the law appropriate to a second-order tensor.

The rate $W$ at which energy is dissipated per unit volume, as a result of the current flow, is given by $\mathbf{E} \cdot \mathbf{j}$. Determine the limits between which $W$ must lie for a given value of $|\mathbf{E}|$ as the direction of $\mathbf{E}$ is varied.

Salamat Ali

Numerade Educator

Problem 15

In a certain system of units, the electromagnetic stress tensor $M_{i j}$ is given by
$$
M_{i j}=E_{i} E_{j}+B_{i} B_{j}-\frac{1}{2} \delta_{i j}\left(E_{k} E_{k}+B_{k} B_{k}\right)
$$
where the electric and magnetic fields, $\mathbf{E}$ and $\mathbf{B}$, are first-order tensors. Show that $M_{i j}$ is a second-order tensor.

Consider a situation in which $|\mathbf{E}|=|\mathbf{B}|$, but the directions of $\mathbf{E}$ and $\mathbf{B}$ are not parallel. Show that $\mathbf{E} \pm \mathbf{B}$ are principal axes of the stress tensor and find the corresponding principal values. Determine the third principal axis and its corresponding principal value.

Raj Bala

Numerade Educator

Problem 16

A rigid body consists of four particles of masses $m, 2 m, 3 m, 4 m$, respectively situated at the points $(a, a, a),(a,-a,-a),(-a, a,-a),(-a,-a, a)$ and connected together by a light framework.
(a) Find the inertia tensor at the origin and show that the principal moments of inertia are $20 m a^{2}$ and $(20 \pm 2 \sqrt{5}) m a^{2}$.(b) Find the principal axes and verify that they are orthogonal.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 17

A rigid body consists of eight particles, each of mass $m$, held together by light rods. In a certain coordinate frame the particles are at positions
$$
\pm a(3,1,-1), \quad \pm a(1,-1,3), \quad \pm a(1,3,-1), \quad \pm a(-1,1,3)
$$
Show that, when the body rotates about an axis through the origin, if the angular velocity and angular momentum vectors are parallel then their ratio must be $40 \mathrm{ma}^{2}, 64 \mathrm{ma}^{2}$ or $72 \mathrm{ma}^{2}$

Keshav Singh

Numerade Educator

Problem 18

The paramagnetic tensor $\chi_{i j}$ of a body placed in a magnetic field, in which its energy density is $-\frac{1}{2} \mu_{0} \mathbf{M} \cdot \mathbf{H}$ with $M_{i}=\sum_{j} \chi_{i j} H_{j}$, is
$$
\left(\begin{array}{ccc}
2 k & 0 & 0 \\
0 & 3 k & k \\
0 & k & 3 k
\end{array}\right)
$$
Assuming depolarizing effects are negligible, find how the body will orientate itself if the field is horizontal, in the following circumstances:
(a) the body can rotate freely;
(b) the body is suspended with the $(1,0,0)$ axis vertical;
(c) the body is suspended with the $(0,1,0)$ axis vertical.

Anand Jangid

Numerade Educator

Problem 19

A block of wood contains a number of thin soft-iron nails (of constant permeability). A unit magnetic field directed eastwards induces a magnetic moment in the block having components $(3,1,-2)$, and similar fields directed northwards and vertically upwards induce moments $(1,3,-2)$ and $(-2,-2,2)$ respectively. Show that all the nails lie in parallel planes.

Ankur S

Numerade Educator

Problem 20

For tin, the conductivity tensor is diagonal, with entries $a, a$, and $b$ when referred to its crystal axes. A single crystal is grown in the shape of a long wire of length $L$ and radius $r$, the axis of the wire making polar angle $\theta$ with respect to the crystal's 3-axis. Show that the resistance of the wire is $L\left(\pi r^{2} a b\right)^{-1}\left(a \cos ^{2} \theta+b \sin ^{2} \theta\right) .$

Ajay Singhal

Numerade Educator

Problem 21

By considering an isotropic body subjected to a uniform hydrostatic pressure (no shearing stress), show that the bulk modulus $k$, defined by the ratio of the pressure to the fractional decrease in volume, is given by $k=E /[3(1-2 \sigma)]$ where $E$ is Young's modulus and $\sigma$ is Poisson's ratio.

Kamlesh Goyal

Numerade Educator

Problem 22

For an isotropic elastic medium under dynamic stress, at time $t$ the displacement $u_{i}$ and the stress tensor $p_{i j}$ satisfy
$$
p_{i j}=c_{i j k l}\left(\frac{\partial u_{k}}{\partial x_{l}}+\frac{\partial u_{l}}{\partial x_{k}}\right) \quad \text { and } \quad \frac{\partial p_{i j}}{\partial x_{j}}=\rho \frac{\partial^{2} u_{i}}{\partial t^{2}}
$$
where $c_{i j k l}$ is the isotropic tensor given in equation $(26.47)$ and $\rho$ is a constant. Show that both $\nabla \cdot \mathbf{u}$ and $\nabla \times \mathbf{u}$ satisfy wave equations and find the corresponding wave speeds.

Dr. Rajveer Singh

Dr. Rajveer Singh

Numerade Educator

Problem 23

A fourth-order tensor $T_{i j k l}$ has the properties
$$
T_{j i k l}=-T_{i j k l}, \quad T_{i j l k}=-T_{i j k l}
$$
Prove that for any such tensor there exists a second-order tensor $K_{m n}$ such that
$$
T_{i j k l}=\epsilon_{i j m} \epsilon_{k l n} K_{m n}
$$
and give an explicit expression for $K_{m n^{*}}$ Consider two (separate) special cases, as follows.
(a) Given that $T_{i j k l}$ is isotropic and $T_{i j j i}=1$, show that $T_{i j k l}$ is uniquely determined and express it in terms of Kronecker deltas.
(b) If now $T_{i j k l}$ has the additional property
$$
T_{k l i j}=-T_{i j k l}
$$
show that $T_{i j k l}$ has only three linearly independent components and find an expression for $T_{i j k l}$ in terms of the vector
$$
V_{i}=-\frac{1}{4} \epsilon_{j k l} T_{i j k l}
$$

Raj Bala

Numerade Educator

Problem 24

Working in cylindrical polar coordinates $\rho, \phi, z$, parameterise the straight line (geodesic) joining $(1,0,0)$ to $(1, \pi / 2,1)$ in terms of $s$, the distance along the line. Show by substitution that the geodesic equations, derived at the end of section 26.22, are satisfied.

Raj Bala

Numerade Educator

Problem 25

In a general coordinate system $u^{i}, i=1,2,3$, in three-dimensional Euclidean space, a volume element is given by
$$
d V=\left|\mathbf{e}_{1} d u^{1} \cdot\left(\mathbf{e}_{2} d u^{2} \times \mathbf{e}_{3} d u^{3}\right)\right|
$$
Show that an alternative form for this expression, written in terms of the determinant $g$ of the metric tensor, is given by
$$
d V=\sqrt{g} d u^{1} d u^{2} d u^{3}
$$
Show that, under a general coordinate transformation to a new coordinate system $u^{\prime \prime}$, the volume element $d V$ remains unchanged, i.e. show that it is a scalar quantity.

Raj Bala

Numerade Educator

Problem 26

By writing down the expression for the square of the infinitesimal arc length $(d s)^{2}$ in spherical polar coordinates, find the components $g_{i j}$ of the metric tensor in this coordinate system. Hence, using $(26.97)$, find the expression for the divergence of a vector field $\mathbf{v}$ in spherical polars. Calculate the Christoffel symbols (of the second kind) $\Gamma_{j k}^{i}$ in this coordinate system.

Jacob Fry

Numerade Educator

Problem 27

Find an expression for the second covariant derivative $v_{i ; j k} \equiv\left(v_{i ; j}\right)_{i k}$ of a vector $v_{i}$ (see $\left.(26.88)\right)$. By interchanging the order of differentiation and then subtracting the two expressions, we define the components $R_{i j k}^{l}$ of the Riemann tensor as
$$
v_{i ; j k}-v_{i ; k j} \equiv R_{i j k}^{l} v_{l}
$$
Show that in a general coordinate system $u^{i}$ these components are given by
$$
R_{i j k}^{l}=\frac{\partial \Gamma_{i k}^{l}}{\partial u^{j}}-\frac{\partial \Gamma^{\prime}{ }_{i j}}{\partial u^{k}}+\Gamma^{m}{ }_{i k} \Gamma_{m j}^{l}-\Gamma_{{ } i j}^{m} \Gamma_{m k}^{l}
$$
By first considering Cartesian coordinates, show that all the components $R_{i j k}^{\prime} \equiv 0$ for any coordinate system in three-dimensional Euclidean space.
In such a space, therefore, we may change the order of the covariant derivatives without changing the resulting expression.

Raj Bala

Numerade Educator

Problem 28

A curve $\mathbf{r}(t)$ is parameterised by a scalar variable $t .$ Show that the length of the curve between two points, $A$ and $B$, is given by
$$
L=\int_{A}^{B} \sqrt{g_{i j} \frac{d u^{i}}{d t} \frac{d u^{\prime}}{d t}} d t
$$
Using the calculus of variations (see chapter 22 ), show that the curve $\mathbf{r}(t)$ that minimises $L$ satisfies the equation
$$
\frac{d^{2} u^{i}}{d t^{2}}+\Gamma^{i}{ }_{j k} \frac{d u^{j}}{d t} \frac{d u^{k}}{d t}=\frac{\ddot{s}}{s} \frac{d u^{i}}{d t}
$$
where $s$ is the arc length along the curve, $s=d s / d t$ and $\mathfrak{s}=d^{2} s / d t^{2} .$ Hence, show that if the parameter $t$ is of the form $t=a s+b$, where $a$ and $b$ are constants, then we recover the equation for a geodesic (26.101).
[A parameter which, like $t$, is the sum of a linear transformation of $s$ and a translation is called an affine parameter.]

Jacob Fry

Numerade Educator

Problem 29

We may define Christoffel symbols of the first kind by
$$
\Gamma_{i j k}=g_{i l} \Gamma_{j k}^{l}
$$
Show that these are given by
$$
\Gamma_{k i j}=\frac{1}{2}\left(\frac{\partial g_{i k}}{\partial u^{j}}+\frac{\partial g_{j k}}{\partial u^{i}}-\frac{\partial g_{i j}}{\partial u^{k}}\right)
$$
By permuting indices, verify that
$$
\frac{\partial g_{i j}}{\partial u^{k}}=\Gamma_{i j k}+\Gamma_{j i k}
$$
Using the fact that $\Gamma_{j k}^{l}=\Gamma_{k j}^{l}$, show that
$$
g_{i j ; k} \equiv 0
$$
i.e. that the covariant derivative of the metric tensor is identically zero in all coordinate systems.

Raj Bala

Numerade Educator