UNIVERSITY OF DUBLIN
XMA2331/2
TRINITY COLLEGE
Faculty of Science
school of mathematics
SF Mathematics
SF Theoretical Physics
SF Two Subject Mod
Trinity Term 2011
Course MA2331 & MA2332: Equations of Mathematical Physics I/II
Thursday, May 19
SPORTS CTR
9:30 – 12:30
Dr. S. Ryan
ATTEMPT SIX QUESTIONS:
THREE FROM SECTION A AND THREE FROM SECTION B
All questions carry equal marks.
Log tables are available from the invigilators, if required.
Non-programmable calculators are permitted for this examination,—please indicate the make
and model of your calculator on each answer book used.
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XMA2331/2
SECTION A
1. Show that the Fourier transform of the cosine function is the sum of two delta
functions.
2
Show by calculating the Fourier transform of f (x) = e−πx , x ∈ R that the Fourier
transform of a Gaussian function is itself a Gaussian function.
Note: you will find the standard Gaussian integral formula in the “Useful formulae”
page.
2. Show that the Jacobian for the transformation from Cartesian to spherical polar
coordinates in two dimensions is J = r.
Evaluate
p the flux of the vector field F = yi − xj + zk through the conic surface
z = x2 + y 2 , 0 ≤ z ≤ 2, oriented upwards.
3. Rewrite the expression for the iterated integral
Z 1 Z 1 Z y
I=
dx
dy
dzf (x, y, z),
0
x
x
with the order of integrations reversed.
Evaluate the integral
Z
1
Z
3
0
2
dxex .
dy
3y
4. Define a conservative vector field and show that a vector field is conservative if and
only if its line integral is path independent.
Evaluate the integral
Z
F · dr,
I=
C
for the vector field F = 3x2 i + 2yzi + y 2 k from (0, 1, 2) to (3, 1, −7) by showing that
F has a (scalar) potential, φ.
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XMA2331/2
SECTION B
5. Solve for the general solution, the following ODEs
(a) y 00 + 2y 0 + 2y = 0 with y(π/4) = 2 and y 0 (π/4) = −2.
(b) y 00 + y = f (x) where f (x) is the periodic square wave defined by
1
0<x<π
f (x) =
−1 −π < x < 0
with f (x + 2π) = f (x).
(c) y 00 (x) + 4y 0 (x) + 4y = e−2x cos(2x).
6. Using separation of variables, or otherwise, solve the two-dimensional Laplace equation
∇2 V (x, y) = 0,
2πx
subject to the boundary conditions: V (0, y) = V (d, y) = 0 and V (x, 0) = V0 sin
,
d
with d and V0 ∈ R .
7. State in full Stokes’ theorem.
Use Stokes’ theorem to evaluate (∇ × F) · dA where the vector field F = z 2 i −
3xyj + x3 y 3 k and S is the surface described by z = 5 − x2 − y 2 above the z = 1
plane, oriented upwards.
Explain why Stokes’ theorem cannot be used to transform the surface integral
Z
(xi + yj + zk) · dA,
S
into a line integral on the boundary of S.
8. Use the generating function
2
Φ(x, h) = e2xh−h =
∞
X
hn
n=0
n!
Hn (x),
to obtain polynomial solutions of Hermite’s equation, y 00 − 2xy 0 + 2αy = 0, for
α = 3, 4 and 5.
Check that your answers agree, up to a multiplicative factor, with the polynomials
arising from the recursion relation
an+2 =
2(n − α)
an .
(n + 1)(n + 2)
4
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XMA2331/2
Some useful formulae
1. A function with period l has a Fourier series expansion
∞
a0 X
f (x) =
+
an cos
2
n=1
2πnx
l
+
∞
X
bn sin
n=1
2πnx
l
where
a0
an
bn
l/2
Z
2
=
l
dxf (x),
−l/2
l/2
2πnx
dxf (x) cos
,
l
−l/2
Z
2πnx
2 l/2
dxf (x) sin
=
.
l −l/2
l
Z
2
=
l
2. A function with period l has a Fourier series expansion
∞
X
f (x) =
cn exp
n=−∞
where
Z
1
cn =
l
l/2
2iπnx
l
dxf (x) exp
−l/2
,
−2iπnx
l
.
3. The Fourier integral representation (or Fourier transform) is
Z ∞
f (x) =
dk fg
(k)eikx ,
−∞
Z ∞
1
g
f (k) =
dxf (x)e−ikx .
2π −∞
4. The Gaussian integral is
Z
∞
−αx2
e
−∞
r
=
π
, α > 0.
α
c UNIVERSITY OF DUBLIN 2011
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