Mathematical Methods for Physics and Engineering: A Comprehensive Guide

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

Chapter 11 Line, surface and volume integrals - all with Video Answers

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Chapter Questions

Problem 1

The vector field $\mathbf{F}$ is defined by
$$
\mathbf{F}=2 x z \mathbf{i}+2 y z^{2} \mathbf{j}+\left(x^{2}+2 y^{2} z-1\right) \mathbf{k}
$$
Calculate $\nabla \times \mathbf{F}$ and deduce that $\mathbf{F}$ can be written $F=\nabla \phi$. Determine the form of $\phi$ ]

Jacob Fry

Numerade Educator

06:58

Problem 2

The vector field $Q$ is defined by
$$
\mathbf{Q}=\left[3 x^{2}(y+z)+y^{3}+z^{3}\right] \mathbf{i}+\left[3 y^{2}(z+x)+z^{3}+x^{3}\right] \mathbf{j}+\left[3 z^{2}(x+y)+x^{3}+y^{3}\right] \mathbf{k}
$$
Show that $Q$ is a conservative field, construct its potential function and hence evaluate the integral $J=\int \mathbf{Q} \cdot d \mathbf{r}$ along any line connecting the point $A$ at $(1,-1,1)$ to $B$ at $(2,1,2)$

Jacob Fry

Numerade Educator

07:34

Problem 3

$\mathbf{F}$ is a vector field $x y^{2} \mathbf{i}+2 \mathbf{j}+x \mathbf{k}$, and $L$ is a path parameterised by $x=c t, y=c / t$, $z=d$ for the range $1 \leq t \leq 2$. Evaluate (a) $\int_{L} \mathbf{F} d t$, (b) $\int_{L} \mathbf{F} d y$ and (c) $\int_{L} \mathbf{F} \cdot d \mathbf{r}$.

Jacob Fry

Numerade Educator

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Problem 4

By making an appropriate choice for the functions $P(x, y)$ and $Q(x, y)$ that appear in Green's theorem in a plane, show that the integral of $x-y$ over the upper half of the unit circle centred on the origin has the value $-\frac{2}{3}$. Show the same result by direct integration in Cartesian coordinates.

Eduard Sanchez

Numerade Educator

08:33

Problem 5

Determine the point of intersection $P$, in the first quadrant, of the two ellipses
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad \text { and } \quad \frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1
$$
Taking $b<a$, consider the contour $L$ that bounds that area in the first quadrant which is common to the two ellipses. Show that the parts of $L$ that lie along the coordinate axes contribute nothing to the line integral around $L$ of $x d y-y d x$, and that this line integral can be written as the sum of two such integrals, $I_{1}$
and $I_{2}$, around closed contours. Using a parameterisation of each ellipse similar to that employed in the example in section $11.3$, evaluate these two integrals and hence find the total area common to the two ellipses.

Jacob Fry

Numerade Educator

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Problem 6

By using parameterisations of the form $x=a \cos ^{n} \theta$ and $y=a \sin ^{n} \theta$ for suitable values of $n$, find the area bounded by the curves
$$
x^{2 / 5}+y^{2 / 5}=a^{2 / 5} \quad \text { and } \quad x^{2 / 3}+y^{2 / 3}=a^{2 / 3}
$$

Eduard Sanchez

Numerade Educator

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Problem 7

Evaluate the line integral
$$
I=\oint_{C}\left[y\left(4 x^{2}+y^{2}\right) d x+x\left(2 x^{2}+3 y^{2}\right) d y\right]
$$
around the ellipse $x^{2} / a^{2}+y^{2} / b^{2}=1$

Eduard Sanchez

Numerade Educator

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Problem 8

Criticise the following 'proof' that $\pi=0$
(a) Apply Green's theorem in a plane to the functions $P(x, y)=\tan ^{-1}(y / x)$ and $Q(x, y)=\tan ^{-1}(x / y)$, taking the region $R$ to be the unit circle centred on the origin.
(b) The RHS of the equality so produced is
$$
\iint_{R} \frac{y-x}{x^{2}+y^{2}} d x d y
$$
which, either by symmetry considerations or by changing to plane polar coordinates, can be shown to have zero value.
(c) In the LHS of the equality set $x=\cos \theta$ and $y=\sin \theta$, yielding $P(\theta)=\theta$ and $Q(\theta)=\pi / 2-\theta .$ The line integral becomes
$$
\int_{0}^{2} \pi\left[\left(\frac{\pi}{2}-\theta\right) \cos \theta-\theta \sin \theta\right] d \theta
$$
which has value $2 \pi$
(d) Thus $2 \pi=0$ and the stated result follows.

Eduard Sanchez

Numerade Educator

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Problem 9

A single-turn coil $C$ of arbitrary shape is placed in a magnetic field $\mathbf{B}$ and carries a current $I$. Show that the couple acting upon the coil can be written as
$$
\mathbf{M}=I \int_{C}(\mathbf{B} \cdot \mathbf{r}) d \mathbf{r}-I \int_{C} \mathbf{B}(\mathbf{r} \cdot d \mathbf{r})
$$
For a planar rectangular coil of sides $2 a$ and $2 b$ placed with its plane vertical and at an angle $\phi$ to a uniform horizontal field $\mathbf{B}$, show that $\mathbf{M}$ is, as expected, $4 a b B I \cos \phi \mathbf{k}$.

Eduard Sanchez

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Problem 10

Find the vector area $\mathrm{S}$ of the curved surface of the hyperboloid of revolution
$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}+z^{2}}{b^{2}}=1
$$
which lies in the region $z>0$ and $a<x<\lambda a$

Eduard Sanchez

Numerade Educator

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Problem 11

An axially symmetric solid body with its axis $A B$ vertical is immersed in an incompressible fluid of density $\rho_{0}$. Use the following method to show that, whatever the shape of the body, for $\rho=\rho(z)$ in cylindrical polars the Archimedean upthrust is, as expected, $\rho_{0} g V$, where $V$ is the volume of the body.

Express the vertical component of the resultant force $\left(-\int p d \mathbf{S}\right.$, where $p$ is the pressure) on the body in terms of an integral; note that $p=-\rho_{0} g z$ and that for an annular surface element of width $d l, \mathbf{n} \cdot \mathbf{n}_{z} d l=-d r$. Integrate by parts and use the fact that $\rho\left(z_{A}\right)=\rho\left(z_{B}\right)=0$

Eduard Sanchez

Numerade Educator

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Problem 12

Show that the expression below is equal to the solid angle subtended by a rectangular aperture of sides $2 a$ and $2 b$ at a point a distance $c$ from the aperture along the normal to its centre:
$$
\Omega=4 \int_{0}^{b} \frac{a c}{\left(y^{2}+c^{2}\right)\left(y^{2}+c^{2}+a^{2}\right)^{1 / 2}} d y
$$
By setting $y=\left(a^{2}+c^{2}\right)^{1 / 2} \tan \phi$, change this integral into the form
$$
\int_{0}^{\phi_{1}} \frac{4 a c \cos \phi}{c^{2}+a^{2} \sin ^{2} \phi} d \phi
$$
where $\tan \phi_{1}=b /\left(a^{2}+c^{2}\right)^{1 / 2}$, and hence show that
$$
\Omega=4 \tan ^{-1}\left[\frac{a b}{c\left(a^{2}+b^{2}+c^{2}\right)^{1 / 2}}\right]
$$

Eduard Sanchez

Numerade Educator

25:30

Problem 13

A vector field $\mathbf{a}$ is given by $-z x r^{-3} \mathbf{i}-z y r^{-3} \mathbf{j}+\left(x^{2}+y^{2}\right) r^{-3} \mathbf{k}$, where $r^{2}=x^{2}+y^{2}+z^{2}$. Establish that the field is conservative (a) by showing $\nabla \times a=0$ and (b) by constructing its potential function $\phi .$

Linda Winkler

Numerade Educator

11:23

Problem 14

A vector field a is given by $\left(z^{2}+2 x y\right) \mathbf{i}+\left(x^{2}+2 y z\right) \mathbf{j}+\left(y^{2}+2 z x\right) \mathbf{k}$. Show that $\mathbf{a}$ is conservative and that the line integral $\int \mathbf{a} \cdot d \mathbf{r}$ along any line joining $(1,1,1)$ and $(1,2,2)$ has the value 11 .

Linda Winkler

Numerade Educator

02:40

Problem 15

A force $\mathbf{F}(\mathbf{r})$ acts on a particle at $\mathbf{r} .$ In which of the following cases can $\mathbf{F}$ be represented in terms of a potential? Where it can, find the potential.
(a) $\mathbf{F}=F_{0}\left[\mathbf{i}-\mathbf{j}-\frac{2 \mid x-y)}{a^{2}} \mathbf{r}\right] \exp \left(-\frac{r^{2}}{a^{2}}\right)$
(b) $\mathbf{F}=\frac{F_{0}}{a}\left[z \mathbf{k}+\frac{\left(x^{2}+y^{2}-a^{2}\right)}{a^{2}} \mathbf{r}\right] \exp \left(-\frac{r^{2}}{a^{2}}\right) ;$
(c) $\mathbf{F}=F_{0}\left[\mathbf{k}+\frac{a(\mathbf{r} \times \mathbf{k})}{r^{2}}\right] .$

Keshav Singh

Numerade Educator

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Problem 16

One of Maxwell's electromagnetic equations states that all magnetic fields $\mathbf{B}$ are solenoidal (i.e. $\nabla \cdot \mathbf{B}=0$ ). Determine whether each of the following vectors could represent a real magnetic field; where it could, try to find a suitable vector potential $\mathbf{A}$, i.e. such that $\mathbf{B}=\nabla \times \mathbf{A}$. (Hint: seek a vector potential that is parallel to $\nabla \times \mathbf{B} .)$
(a) $\frac{B_{0} b}{r^{3}}\left[(x-y) z \mathbf{i}+(x-y) z \mathbf{j}+x^{2}-y^{2} \mathbf{k}\right]$ in Cartesians with $r^{2}=x^{2}+y^{2}+z^{2}$
(b) $\frac{B_{0} b}{r^{3}}\left[\cos \theta \cos \phi \hat{\mathbf{e}}_{r}-\sin \theta \cos \phi \hat{\mathbf{e}}_{\theta}+\sin 2 \theta \sin \phi \hat{\mathbf{e}}_{\phi}\right]$ in spherical polars;
(c) $B_{0} b^{2}\left[\frac{z r}{\left(b^{2}+z^{2}\right)^{2}} \hat{\mathbf{e}}_{\rho}+\frac{1}{b^{2}+z^{2}} \hat{\mathbf{e}}_{z}\right]$ in cylindrical polars.

Eduard Sanchez

Numerade Educator

07:34

Problem 17

The vector field $\mathbf{f}$ has components $y \mathbf{i}-x \mathbf{j}+\mathbf{k}$ and $\gamma$ is a curve given parametrically by
$$
\mathbf{r}=(a-c+c \cos \theta) \mathbf{i}+(b+c \sin \theta) \mathbf{j}+c^{2} \theta \mathbf{k}, \quad 0 \leq \theta \leq 2 \pi
$$
Describe the shape of the path $\gamma$ and show that the line integral $\int_{y} \mathbf{f} \cdot d \mathbf{r}$ vanishes. Does this result imply that $\mathbf{f}$ is a conservative field?

Jacob Fry

Numerade Educator

06:53

Problem 18

A vector field $\mathbf{a}=f(r) \mathbf{r}$ is spherically symmetric and everywhere directed away from the origin. Show that a is irrotational but that it is also solenoidal only if $f(r)$ is of the form $A r^{-3}$.

Linda Winkler

Numerade Educator

02:40

Problem 19

Evaluate the surface integral $\int \mathbf{r} \cdot d \mathbf{S}$, where $\mathbf{r}$ is the position vector, over that part of the surface $z=a^{2}-x^{2}-y^{2}$ for which $z \geq 0$, by each of the following methods:
(a) parameterize the surface as $x=a \sin \theta \cos \phi, y=a \sin \theta \sin \phi, z=a^{2} \cos ^{2} \theta$, and show that
$$
\mathbf{r} \cdot d \mathbf{S}=a^{4}\left(2 \sin ^{3} \theta \cos \theta+\cos ^{3} \theta \sin \theta\right) d \theta d \phi
$$
(b) apply the divergence theorem to the volume bounded by the surface and the plane $z=0$

Nick Johnson

Numerade Educator

02:00

Problem 20

Obtain an expression for the value $\phi_{P}$ at a point $P$ of a scalar function $\phi$ that satisfies $\nabla^{2} \phi=0$ in terms of its value and normal derivative on a surface $S$ that encloses it, by proceeding as follows.
(a) In Green's second theorem take $\psi$ at any particular point $Q$ as $1 / r$, where $r$ is the distance of $Q$ from $P$. Show that $\nabla^{2} \varphi=0$ except at $r=0$
(b) Apply the result to the doubly connected region bounded by $S$ and a small sphere $\Sigma$ of radius $\delta$ centred on $\mathrm{P}$.
(c) Apply the divergence theorem to show that the surface integral over $\Sigma$ involving $1 / \delta$ vanishes, and prove that the term involving $1 / \delta^{2}$ has the value $4 \pi \phi_{P}$
(d) Conclude that
$$
\phi p=-\frac{1}{4 \pi} \int_{S} \phi \frac{\partial}{\partial n}\left(\frac{1}{r}\right) d S+\frac{1}{4 \pi} \int_{S} \frac{1}{r} \frac{\partial \phi}{\partial n} d S
$$
This important result shows that the value at a point $P$ of a function $\phi$ that satisfies $\nabla^{2} \phi=0$ everywhere within a closed surface $S$ that encloses $P$ may be expressed entirely in terms of its value and normal derivative on $S .$ This matter is taken up more generally in connection with Green's functions in chapter 19 and in connection with functions of a complex variable in section $20.12$

Dorcas Attuabea Addo

Numerade Educator

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Problem 21

Use result $(11.21)$, together with an appropriately chosen scalar function $\phi$ to prove that the position vector $\overline{\mathbf{r}}$ of the centre of mass of an arbitrarily-shaped body of volume $V$ and uniform density can be written
$$
\overline{\mathbf{r}}=\frac{1}{V} \oint_{S} \frac{1}{2} r^{2} d \mathbf{S}
$$

Eduard Sanchez

Numerade Educator

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Problem 22

A rigid body of volume $V$ and surface $S$ rotates with angular velocity $\omega .$ Show that
$$
\omega=-\frac{1}{2 V} \oint_{S} \mathbf{u} \times d \mathbf{S}
$$
where $\mathbf{u}(\mathbf{x})$ is the velocity of the point $\mathbf{x}$ on the surface $S$.

Eduard Sanchez

Numerade Educator

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Problem 23

Demonstrate the validity of the divergence theorem:
(a) by calculating the flux of the vector
$$
\mathbf{F}=\frac{\alpha \mathbf{r}}{\left(r^{2}+a^{2}\right)^{3 / 2}}
$$
through the spherical surface $|\mathbf{r}|=\sqrt{3} a$
(b) by showing that
$$
\nabla \cdot \mathbf{F}=\frac{3 x a^{2}}{\left(r^{2}+a^{2}\right)^{5 / 2}}
$$
and evaluating the volume integral of $\nabla \cdot \mathbf{F}$ over the interior of the sphere $|\mathbf{r}|=\sqrt{3} a$
(The substitution $r=a \tan \theta$ will prove useful in carrying out the integration.)

Eduard Sanchez

Numerade Educator

04:44

Problem 24

Prove equation (11.22) and, by taking $\mathbf{b}=z x^{2} \mathbf{i}+z y^{2} \mathbf{j}+\left(x^{2}-y^{2}\right) \mathbf{k}$, show that the two integrals
$$
I=\int x^{2} d V \text { and } J=\int \cos ^{2} \theta \sin ^{3} \theta \cos 2 \phi d \theta d \phi
$$
both taken over the unit sphere, must have the same value. Evaluate both directly to show that the common value is $4 \pi / 15$.

Sriram Soundarrajan

Numerade Educator

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Problem 25

In a uniform, non-dielectric, conducting medium with unit relative permittivity, charge density $\rho$, current density $\mathbf{J}$, electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$, Maxwell's electromagnetic equations take the form (with $\left.\mu_{0} \epsilon_{0}=c^{-2}\right)$
(i) $\nabla \cdot \mathbf{B}=0$,
(ii) $\nabla \cdot \mathbf{E}=\rho / \epsilon_{0}$,
(iii) $\nabla \times \mathbf{E}+\mathbf{B}=\mathbf{0}$,
(iv) $\nabla \times \mathbf{B}-\left(\hat{\mathbf{E}} / c^{2}\right)=\mu_{0} \mathbf{J}$,
The density of stored energy in the medium is given by $\frac{1}{2}\left(\epsilon_{0} E^{2}+\mu_{0}^{-1} B^{2}\right)$. Show that the rate of change of the total stored energy in a volume $V$ is equal to
$$
-\int_{V} \mathbf{J} \cdot \mathbf{E} d V-\frac{1}{\mu_{0}} \oint_{S}(\mathbf{E} \times \mathbf{B}) \cdot d \mathbf{S}
$$
where $S$ is the surface bounding $V$. (The first integral gives the ohmic heating loss, whilst the second gives the electromagnetic energy flux out of the bounding surface. The vector $\mu_{0}^{-1}(\mathbf{E} \times \mathbf{B})$ is known as the Poynting vector.)

Eduard Sanchez

Numerade Educator

02:13

Problem 26

A vector field $\mathbf{F}$ is defined in cylindrical polar coordinates $\rho, \theta, z$ by
$$
\mathbf{F}=F_{0}\left(\frac{x \cos \lambda z}{a} \mathbf{i}+\frac{y \cos \lambda z}{a} \mathbf{j}+(\sin \lambda z) \mathbf{k}\right) \equiv \frac{\rho}{a}(\cos \lambda z) \mathbf{e}_{\rho}+(\sin \lambda z) \mathbf{k}
$$
where $\mathbf{i}, \mathbf{j}$ and $\mathbf{k}$ are the unit vectors along the Cartesian axes and $\mathrm{e}_{\rho}$ is the unit vector $(x / \rho) \mathbf{i}+(y / \rho) \mathbf{j}$
(a) Calculate, as a surface integral, the flux of $\mathbf{F}$ through the closed surface bounded by the cylinders $\rho=a$ and $\rho=2 a$ and the planes $z=\pm a \pi / 2$
(b) Evaluate the same integral using the divergence theorem.

Joseph Liao

Numerade Educator

06:58

Problem 27

The vector field $\mathbf{F}$ is given by
$$
\mathbf{F}=\left(3 x^{2} y z+y^{3} z+x e^{-x}\right) \mathbf{i}+\left(3 x y^{2} z+x^{3} z+y e^{x}\right) \mathbf{j}+\left(x^{3} y+y^{3} x+x y^{2} z^{2}\right) \mathbf{k}
$$
Calculate (a) directly and (b) by using Stokes' theorem the value of the line integral $\int_{L} \mathbf{F} \cdot d \mathbf{r}$, where $L$ is the (three-dimensional) closed contour $O A B C D E O$ defined by the successive vertices $(0,0,0),(1,0,0),(1,0,1),(1,1,1),(1,1,0),(0,1,0)$, $(0,0,0)$

Jacob Fry

Numerade Educator

17:08

Problem 28

A vector force field $\mathbf{F}$ is defined in Cartesian coordinates by
$$
\mathbf{F}=F_{0}\left[\left(\frac{y^{3}}{3 a^{3}}+\frac{y}{a} e^{x y / a^{2}}+1\right) \mathbf{i}+\left(\frac{x y^{2}}{a^{3}}+\frac{x+y}{a} e^{x y / a^{2}}\right) \mathbf{j}+\frac{z}{a} e^{x y / a^{2}} \mathbf{k}\right]
$$
Use Stokes' theorem to calculate
$$
\oint_{L} \mathbf{F} \cdot d \mathbf{r}
$$
where $L$ is the perimeter of the rectangle $A B C D$ given by $A=(0,1,0), B=(1,1,0)$, $C=(1,3,0)$ and $D=(0,3,0)$

Linda Winkler

Numerade Educator