Mathematical Methods for Physics and Engineering: A Comprehensive Guide

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

Chapter 5 Partial differentiation - all with Video Answers

Educators

Chapter Questions

Problem 1

(a) Find all the first partial derivatives of the following functions $f(x, y)$ : (i) $x^{2} y$,
(ii) $x^{2}+y^{2}+4$, (iii) $\sin (x / y)$, (iv) $\tan ^{-1}(y / x)$, (v) $r(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}$.
182(b) For (i), (ii) and (v), find $\partial^{2} f / \partial x^{2}, \partial^{2} f / \partial y^{2}, \partial^{2} f / \partial x \partial y$.
(c) For (iv) verify that $\partial^{2} f / \partial x \partial y=\partial^{2} f / \partial y \partial x$.

Foster Wisusik

Numerade Educator

00:56

Problem 3

Show that the differential
$$
d f=x^{2} d y-\left(y^{2}+x y\right) d x
$$
is not exact, but that $d g=\left(x y^{2}\right)^{-1} d f$ is exact.
(a) Show that
$$
d f=y\left(1+x-x^{2}\right) d x+x(x+1) d y
$$
is not an exact differential.
(b) Find the differential equation that a function $g(x)$ must satisfy if $d \phi=g(x) d f$ is to be an exact differential. Verify that $g(x)=e^{-x}$ is a solution of this equation and deduce the form of $\phi(x, y) .$

Vikash Ranjan

Numerade Educator

02:15

Problem 4

(a) Show that
$$
d f=y\left(1+x-x^{2}\right) d x+x(x+1) d y
$$
is not an exact differential.
(b) Find the differential equation that a function $g(x)$ must satisfy if $d \phi=g(x) d f$ is to be an exact differential. Verify that $g(x)=e^{-x}$ is a solution of this equation and deduce the form of $\phi(x, y)$.

Umar Sohail Qureshi

Numerade Educator

02:57

Problem 5

The equation $3 y=z^{3}+3 x z$ defines $z$ implicitly as a function of $x$ and $y .$ Evaluate all three second partial derivatives of $z$ with respect to $x$ and/or $y$. Verify that $z$. is a solution of
$$
x \frac{\partial^{2} z}{\partial y^{2}}+\frac{\hat{\partial}^{2} z}{\partial x^{2}}=0
$$

Shivani Yadav

Numerade Educator

04:16

Problem 6

A possible equation of state for a gas takes the form
$$
p V=R T \exp \left(-\frac{\alpha}{V R T}\right)
$$
in which $\alpha$ and $R$ are constants. Calculate expressions for
$$
\left(\frac{\partial p}{\partial V}\right)_{T}, \quad\left(\frac{\partial V}{\partial T}\right)_{p}, \quad\left(\frac{\partial T}{\partial p}\right)_{V}
$$
and show that their product is $-1$, as stated in section $5.4$

Nick Derr

Numerade Educator

00:54

Problem 7

The function $G(t)$ is defined by
$$
G(t)=F(x, y)=x^{2}+y^{2}+3 x y
$$
where $x(t)=a t^{2}$ and $y(t)=2 a t$. Use the chain rule to find the values of $(x, y)$ at which $G(t)$ has stationary values as a function of $t$. Do any of them correspond, to the stationary points of $F(x, y)$ as a function of $x$ and $y$ ?

Linda Hand

Numerade Educator

04:49

Problem 8

In the $x y$-plane, new coordinates $s$ and $t$ are defined by
$$
s=\frac{1}{2}(x+y), \quad t=\frac{1}{2}(x-y)
$$
Transform the equation
$$
\frac{\partial^{2} \phi}{\partial x^{2}}-\frac{\partial^{2} \phi}{\partial y^{2}}=0
$$
into the new coordinates and deduce that its general solution can be written
$$
\phi(x, y)=f(x+y)+g(x-y)
$$
where $f(u)$ and $g(v)$ are arbitrary functions of $u$ and $v$ respectively.
183

Charles Machakwa

Numerade Educator

05:23

Problem 9

The function $f(x, y)$ satisfies the differential equation
$$
y \frac{\partial f}{\partial x}+x \frac{\partial f}{\partial y}=0
$$
By changing to new variables $u=x^{2}-y^{2}$ and $v=2 x y$, show that $f$ is, in fact, a function of $x^{2}-y^{2}$ only.

Samuel Smith

Numerade Educator

13:39

Problem 10

If $x=e^{\mu} \cos \theta$ and $y=e^{u} \sin \theta$, show that
$$
\frac{\partial^{2} \phi}{\partial u^{2}}+\frac{\hat{\partial}^{2} \phi}{\partial \theta^{2}}=\left(x^{2}+y^{2}\right)\left(\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}\right)
$$
where $f(x, y)=\phi(u, \theta)$.

Samuel Smith

Numerade Educator

09:00

Problem 11

Find and evaluate the maxima, minima and saddle points of the function
$$
f(x, y)=x y\left(x^{2}+y^{2}-1\right)
$$

Linda Hand

Numerade Educator

02:19

Problem 12

Show that
$$
f(x, y)=x^{3}-12 x y+48 x+b y^{2}, \quad b \neq 0
$$
has two, one, or zero stationary points according to whether $|b|$ is less than, equal to, or greater than $3 .$

Sriram Soundarrajan

Numerade Educator

08:36

Problem 13

Locate the stationary points of the function
$$
f(x, y)=\left(x^{2}-2 y^{2}\right) \exp \left[-\left(x^{2}+y^{2}\right) / a^{2}\right]
$$
where $a$ is a non-zero constant.
Sketch the function along the $x$ - and $y$-axes and hence identify the nature and values of the stationary points.

Samuel Smith

Numerade Educator

04:07

Problem 14

Find the stationary values of
$$
f(x, y)=4 x^{2}+4 y^{2}+x^{4}-6 x^{2} y^{2}+y^{4}
$$
and classify them as maxima, minima or saddle points. Make a rough sketch of the contours of $f$ in the quarter plane $x, y \geq 0$.

Linh Vu

Numerade Educator

04:07

Problem 15

Linh Vu

Numerade Educator

01:29

Problem 16

The temperature of a point $(x, y, z)$ on the unit sphere is given by
$$
T(x, y, z)=1+x y+y z
$$
By using the method of Lagrange multipliers find the temperature of the hottest point on the sphere.

James Kiss

Numerade Educator

01:47

Problem 17

A rectangular parallelepiped has all eight vertices on the ellipsoid
$$
x^{2}+3 y^{2}+3 z^{2}=1
$$
Using the symmetry of the parallelepiped about each of the planes $x=0$ $y=0, z=0$, write down the surface area of the parallelepiped in terms of the coordinates of the vertex that lies in the octant $x, y, z \geq 0$. Hence find the maximum value of the surface area of such a parallelepiped.

Nick Johnson

Numerade Educator

06:49

Problem 18

Two horizontal corridors, $0 \leq x \leq a$ with $y \geq 0$, and $0 \leq y \leq b$ with $x \geq 0$, meet at right angles. Find the length $L$ of the longest ladder (considered as a stick) that may be carried horizontally around the corner.

David Mccaslin

Numerade Educator

01:34

Problem 19

A barn is to be constructed with a uniform cross-sectional area $A$ throughout its length. The cross-section is to be a rectangle of wall height $h$ (fixed) and width $w$, surmounted by an isosceles triangular roof that makes an angle $\theta$ with

Jay Patel

Numerade Educator

07:07

Problem 20

Show that the envelope of all concentric ellipses that have their axes along the $x$ - and $y$-coordinate axes and that have the sum of their semi-axes equal to a constant $L$ is the same curve (an astroid) as that found in the worked example in section $5.10$.

Harshita Goel

Numerade Educator

09:07

Problem 21

Find the area of the region covered by points on the lines
$$
\frac{x}{a}+\frac{y}{b}=1
$$
where the sum of any line's intercepts on the coordinate axes is fixed and equal to $c$.

Harmender Singh Yadav

Numerade Educator

01:57

Problem 23

A water feature contains a spray head at water level at the centre of a round basin. The head is in the form of a hemisphere with many evenly distributed small holes in it, and through which water spurts out at the same speed $t_{0}$ in all directions.
(a) What is the shape of the 'water bell' so formed?
(b) What must be the minimum diameter of the bowl if no water is to be lost?

Mahipal Kumawat

Numerade Educator

07:39

Problem 24

In order to make a focussing mirror that concentrates parallel axial rays to one spot (or conversely forms a parallel beam from a point source) a parabolic shape should be adopted. If a mirror that is part of a circular cylinder or sphere were used, the light would be spread out along a curve. This curve is known as a caustic and is the envelope of the rays reflected from the mirror. Denoting by $\theta$ the angle which a typical incident axial ray makes with the normal to the mirror at the place where it is reflected, the geometry of reflection (the angle of incidence equals the angle of reflection) is shown in figure $5.5 .$
Show that a parametric specification of the caustic is
$$
x=R \cos \theta\left(\frac{1}{2}+\sin ^{2} \theta\right), \quad y=R \sin ^{3} \theta
$$
where $R$ is the radius of curvature of the mirror. The curve is, in fact, part of an' epicycloid.

Sheh Lit Chang

University of Washington

07:39

Problem 25

Sheh Lit Chang

University of Washington

00:29

Problem 26

Functions $P(V, T), U(V, T)$ and $S(V, T)$ are related by
$$
T d S=d U+P d V
$$
where the symbols have the same meaning as in the previous question. $P$ is known from experiment to have the form,
$$
P=\frac{T^{4}}{3}+\frac{T}{V}
$$
in appropriate units. If
$$
U=\alpha V T^{4}+\beta T
$$
where $\alpha, \beta$, are constants (or at least do not depend on $T, V)$, deduce that $\alpha$ must have a specific value but $\beta$ may have any value. Find the corresponding form of $S$.

Alison Rodriguez

Numerade Educator

01:02

Problem 27

As in the previous two exercises on the thermodynamics of a simple gas, the quantity $d S=T^{-1}(d U+P d V)$ is an exact differential. Use this to prove that
$$
\left(\frac{\partial U}{\partial V}\right)_{T}=T\left(\frac{\partial P}{\partial T}\right)_{V}-P
$$
In the van der Waals model of a gas, $P$ obeys the equation
$$
P=\frac{R T}{V-b}-\frac{a}{V^{2}}
$$
where $R, a$ and $b$ are constants. Further, in the limit $V \rightarrow \infty$, the form of $U$ becomes $U=c T$, where $c$ is another constant. Find the complete expression for $U(V, T)$

Adrian Co

Numerade Educator

15:00

Problem 28

The entropy $S(H, T)$, the magnetisation $M(H, T)$ and the internal energy $U(H, T)$ of a magnetic salt placed in a magnetic field of strength $H$ at temperature $T$ are connected by the equation
$$
\begin{gathered}
T d S=d U-H d M \\
186
\end{gathered}
$$By considering $d(U-T S-H M)$, or otherwise, prove that
$$
\left(\frac{\partial M}{\partial T}\right)_{H}=\left(\frac{\partial S}{\partial H}\right)_{T}
$$
For a particular salt
$$
M(H, T)=M_{0}[1-\exp (-\alpha H / T)]
$$
Show that, at a fixed temperature, if the applied field is increased from zero to a strength such that the magnetization of the salt is $\frac{3}{4} M_{0}$ then the salt's entropy decreases by an amount
$$
\frac{M_{0}}{\Delta c}(3-\ln 4)
$$

Ravindra Yadav

Numerade Educator

07:15

Problem 29

Using the results of section $5.12$, evaluate the integral
$$
I(y)=\int_{0}^{\infty} \frac{e^{-x y} \sin x}{x} d x
$$
Hence show that
$$
J=\int_{0}^{\infty} \frac{\sin x}{x} d x=\frac{\pi}{2}
$$

Dorcas Attuabea Addo

Numerade Educator

03:03

Problem 30

The integral
$$
\int_{-\infty}^{\infty} e^{-x x^{2}} d x
$$
has the value $(\pi / \alpha)^{1 / 2}$. Use this result to evaluate
$$
J(n)=\int_{-\infty}^{\infty} x^{2 n} e^{-x^{2}} d x
$$
where $n$ is a positive integer. Express your answer in terms of factorials.

Mutahar Mehkri

Numerade Educator

01:54

Problem 31

The function $f(x)$ is differentiable and $f(0)=0$. A second function $g(y)$ is defined by
$$
g(y)=\int_{0}^{y} \frac{f(x) d x}{\sqrt{y-x}}
$$
Prove that
$$
\frac{d g}{d y}=\int_{0}^{y} \frac{d f}{d x} \frac{d x}{\sqrt{y-x}}
$$
For the case $f(x)=x^{n}$, prove that
$$
\frac{d^{n} g}{d y^{n}}=2(n !) \sqrt{y}
$$

Anthony Ramos

Numerade Educator

04:37

Problem 32

The functions $f(x, t)$ and $F(x)$ are defined by
$$
\begin{aligned}
f(x, t) &=e^{-x t} \\
F(x) &=\int_{0}^{x} f(x, t) d t
\end{aligned}
$$
Verify by explicit calculation that
$$
\frac{d F}{d x}=f(x, x)+\int_{0}^{x} \frac{\partial f(x, t)}{\partial x} d t
$$
187If
$$
I(\alpha)=\int_{0}^{1} \frac{x^{\alpha}-1}{\ln x} d x, \quad \alpha>-1
$$
what is the value of $I(0)$ ? Show that
$$
\frac{d}{d x} x^{\alpha}=x^{2} \ln x
$$
and deduce that
$$
\frac{d}{d \alpha} I(\alpha)=\frac{1}{\alpha+1}
$$
Hence prove that $I(\alpha)=\ln (1+\alpha)$.

Jeff Vermeire

Numerade Educator

05:23

Problem 34

Find the derivative with respect to $x$ of the integral
$$
I(x)=\int_{x}^{3 x} \exp x t d t
$$

Jeff Vermeire

Numerade Educator

07:17

Problem 35

The function $G(t, \xi)$ is defined for $0 \leq t \leq \pi$ by
$$
G(t, \xi)= \begin{cases}-\cos t \sin \xi & \text { for } \xi \leq t \\ -\sin t \cos \xi & \text { for } \xi>t\end{cases}
$$
Show that the function $x(t)$ defined by
$$
x(t)=\int_{0}^{\pi} G(t, \xi) f(\xi) d \xi
$$
satisfies the equation
$$
\frac{d^{2} x}{d t^{2}}+x=f(t)
$$
for any arbitrary (continuous) function $f(t) .$ Show further that $x(0)=$ $[d x / d t]_{x-x}=0$, again for any $f(t)$, but that the value of $x(\pi)$ does depend upon the form of $f(t)$
(The function $G(t, \bar{\xi})$ is an example of a Green's function, an important concept in the solution of differential equations and one studied extensively in later chapters.)

Jeff Vermeire

Numerade Educator