Exercises

5.13 EXERCISES
constant limits of integration the order of integration and differentiation can be
reversed.
In the more general case where the limits of the integral are themselves functions
of x, it follows immediately that
t=v(x)
f(x, t) dt
I(x) =
t=u(x)
= F(x, v(x)) − F(x, u(x)),
which yields the partial derivatives
∂I
= f(x, v(x)),
∂v
Consequently
dI
=
dx
∂I
∂v
dv
+
dx
∂I
∂u
∂I
= −f(x, u(x)).
∂u
du
∂I
+
dx ∂x
v(x)
du
∂
dv
− f(x, u(x))
+
f(x, t)dt
dx
dx ∂x u(x)
v(x)
du
∂f(x, t)
dv
− f(x, u(x))
+
dt,
= f(x, v(x))
dx
dx
∂x
u(x)
= f(x, v(x))
(5.47)
where the partial derivative with respect to x in the last term has been taken
inside the integral sign using (5.46). This procedure is valid because u(x) and v(x)
are being held constant in this term.
Find the derivative with respect to x of the integral
x2
sin xt
I(x) =
dt.
t
x
Applying (5.47), we see that
x2
sin x3
sin x2
dI
t cos xt
=
(1) +
dt
(2x) −
2
dx
x
x
t
x
x2
2 sin x3
sin x2
sin xt
−
+
=
x
x
x
x
sin x3
sin x2
−2
x
x
1
= (3 sin x3 − 2 sin x2 ). x
=3
5.13 Exercises
5.1
Using the appropriate properties of ordinary derivatives, perform the following.
179
PARTIAL DIFFERENTIATION
(a) Find all the first partial derivatives of the following functions f(x, y):
(i) x2 y, (ii) x2 + y 2 + 4, (iii) sin(x/y), (iv) tan−1 (y/x),
(v) r(x, y, z) = (x2 + y 2 + z 2 )1/2 .
(b) For (i), (ii) and (v), find ∂2 f/∂x2 , ∂2 f/∂y 2 and ∂2 f/∂x∂y.
(c) For (iv) verify that ∂2 f/∂x∂y = ∂2 f/∂y∂x.
5.2
Determine which of the following are exact differentials:
(a)
(b)
(c)
(d)
(e)
5.3
(3x + 2)y dx + x(x + 1) dy;
y tan x dx + x tan y dy;
y 2 (ln x + 1) dx + 2xy ln x dy;
y 2 (ln x + 1) dy + 2xy ln x dx;
[x/(x2 + y 2 )] dy − [y/(x2 + y 2 )] dx.
Show that the differential
df = x2 dy − (y 2 + xy) dx
5.4
is not exact, but that dg = (xy 2 )−1 df is exact.
Show that
df = y(1 + x − x2 ) dx + x(x + 1) dy
5.5
is not an exact differential.
Find the differential equation that a function g(x) must satisfy if dφ = g(x)df
is to be an exact differential. Verify that g(x) = e−x is a solution of this equation
and deduce the form of φ(x, y).
The equation 3y = z 3 + 3xz 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
∂2 z
∂2 z
x 2 + 2 = 0.
∂y
∂x
5.6
5.7
A possible equation of state for a gas takes the form
α ,
P V = RT exp −
V RT
in which α and R are constants. Calculate expressions for
∂P
∂V
∂T
,
,
,
∂V T
∂T P
∂P V
and show that their product is −1, as stated in section 5.4.
The function G(t) is defined by
G(t) = F(x, y) = x2 + y 2 + 3xy,
2
5.8
where x(t) = at and y(t) = 2at. 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?
In the xy-plane, new coordinates s and t are defined by
s = 12 (x + y),
t = 12 (x − y).
Transform the equation
∂2 φ ∂2 φ
− 2 =0
∂x2
∂y
into the new coordinates and deduce that its general solution can be written
φ(x, y) = f(x + y) + g(x − y),
where f(u) and g(v) are arbitrary functions of u and v, respectively.
180
5.13 EXERCISES
5.9
The function f(x, y) satisfies the differential equation
y
5.10
5.11
∂f
∂f
+x
= 0.
∂x
∂y
By changing to new variables u = x2 − y 2 and v = 2xy, show that f is, in fact, a
function of x2 − y 2 only.
If x = eu cos θ and y = eu sin θ, show that
2
∂ f
∂2 f
∂2 φ ∂2 φ
+ 2 = (x2 + y 2 )
+ 2 ,
2
2
∂u
∂θ
∂x
∂y
where f(x, y) = φ(u, θ).
Find and evaluate the maxima, minima and saddle points of the function
f(x, y) = xy(x2 + y 2 − 1).
5.12
Show that
f(x, y) = x3 − 12xy + 48x + by 2 ,
5.13
b = 0,
has two, one, or zero stationary points, according to whether |b| is less than,
equal to, or greater than 3.
Locate the stationary points of the function
f(x, y) = (x2 − 2y 2 ) exp[−(x2 + y 2 )/a2 ],
5.14
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.
Find the stationary points of the function
f(x, y) = x3 + xy 2 − 12x − y 2
5.15
and identify their natures.
Find the stationary values of
f(x, y) = 4x2 + 4y 2 + x4 − 6x2 y 2 + y 4
5.16
and classify them as maxima, minima or saddle points. Make a rough sketch of
the contours of f in the quarter plane x, y ≥ 0.
The temperature of a point (x, y, z) on the unit sphere is given by
5.17
By using the method of Lagrange multipliers, find the temperature of the hottest
point on the sphere.
A rectangular parallelepiped has all eight vertices on the ellipsoid
T (x, y, z) = 1 + xy + yz.
x2 + 3y 2 + 3z 2 = 1.
5.18
5.19
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 ≥ 0. Hence find the
maximum value of the surface area of such a parallelepiped.
Two horizontal corridors, 0 ≤ x ≤ a with y ≥ 0, and 0 ≤ y ≤ b with x ≥ 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.
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 θ with
181
PARTIAL DIFFERENTIATION
the horizontal. The cost of construction is α per unit height of wall and β per
unit (slope) length of roof. Show that, irrespective of the values of α and β, to
minimise costs w should be chosen to satisfy the equation
w 4 = 16A(A − wh),
5.20
5.21
5.22
and θ made such that 2 tan 2θ = w/h.
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.
Find the area of the region covered by points on the lines
x y
+ = 1,
a
b
where the sum of any line’s intercepts on the coordinate axes is fixed and equal
to c.
Prove that the envelope of the circles whose diameters are those chords of a
given circle that pass through a fixed point on its circumference, is the cardioid
r = a(1 + cos θ).
5.23
Here a is the radius of the given circle and (r, θ) are the polar coordinates of the
envelope. Take as the system parameter the angle φ between a chord and the
polar axis from which θ is measured.
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 small hemisphere perforated by many evenly
distributed small holes, through which water spurts out at the same speed, v0 , 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?
5.24
5.25
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 θ
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 θ 12 + sin2 θ ,
y = R sin3 θ,
where R is the radius of curvature of the mirror. The curve is, in fact, part of an
epicycloid.
By considering the differential
dG = d(U + P V − ST ),
where G is the Gibbs free energy, P the pressure, V the volume, S the entropy
and T the temperature of a system, and given further that the internal energy U
satisfies
dU = T dS − P dV ,
derive a Maxwell relation connecting (∂V /∂T )P and (∂S/∂P )T .
182
5.13 EXERCISES
y
θ
R
θ
2θ
O
x
Figure 5.5 The reflecting mirror discussed in exercise 5.24.
5.26
Functions P (V , T ), U(V , T ) and S(V , T ) are related by
T dS = dU + P dV ,
where the symbols have the same meaning as in the previous question. The
pressure P is known from experiment to have the form
P =
T
T4
+ ,
3
V
in appropriate units. If
U = αV T 4 + βT ,
5.27
where α, β, are constants (or, at least, do not depend on T or V ), deduce that α
must have a specific value, but that β may have any value. Find the corresponding
form of S.
As in the previous two exercises on the thermodynamics of a simple gas, the
quantity dS = T −1 (dU + P dV ) is an exact differential. Use this to prove that
∂U
∂P
=T
− P.
∂V T
∂T V
In the van der Waals model of a gas, P obeys the equation
P =
5.28
RT
a
,
−
V − b V2
where R, a and b are constants. Further, in the limit V → ∞, the form of U
becomes U = cT , where c is another constant. Find the complete expression for
U(V , T ).
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
T dS = dU − HdM.
183
PARTIAL DIFFERENTIATION
By considering d(U − T S − HM) prove that
∂M
∂S
=
.
∂T H
∂H T
For a particular salt,
M(H, T ) = M0 [1 − exp(−αH/T )].
Show that if, at a fixed temperature, the applied field is increased from zero to a
strength such that the magnetization of the salt is 34 M0 , then the salt’s entropy
decreases by an amount
M0
(3 − ln 4).
4α
5.29
Using the results of section 5.12, evaluate the integral
∞ −xy
e sin x
dx.
I(y) =
x
0
Hence show that
∞
π
sin x
dx = .
x
2
J=
0
5.30
The integral
∞
e−αx dx
2
−∞
has the value (π/α)1/2 . Use this result to evaluate
∞
2
J(n) =
x2n e−x dx,
−∞
5.31
where n is a positive integer. Express your answer in terms of factorials.
The function f(x) is differentiable and f(0) = 0. A second function g(y) is defined
by
y
f(x) dx
√
g(y) =
.
y−x
0
Prove that
dg
=
dy
y
0
df dx
.
√
dx y − x
For the case f(x) = xn , prove that
√
dn g
= 2(n!) y.
dy n
5.32
The functions f(x, t) and F(x) are defined by
f(x, t) = e−xt ,
x
f(x, t) dt.
F(x) =
0
Verify, by explicit calculation, that
dF
= f(x, x) +
dx
184
x
0
∂f(x, t)
dt.
∂x