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NATURAL SCIENCES TRIPOS Part IB & II (General)
Friday 31 May, 2002 9 to 12
MATHEMATICS (2)
Before you begin read these instructions carefully:
You may submit answers to no more than six questions. All questions carry the same number of marks.
The approximate number of marks allocated to a part of a question is indicated in the right hand margin.
Write on one side of the paper only and begin each answer on a separate sheet.
At the end of the examination:
Each question has a number and a letter (for example, 6B).
Answers must be tied up in separate bundles, marked A, B or C according to the letter affixed to each question.
Do not join the bundles together.
For each bundle, a blue cover sheet must be completed and attached to the bundle.
A separate yellow master cover sheet listing all the questions attempted must also be completed.
Every cover sheet must bear your examination number and desk number.
1A A slow viscous flow in two dimensions is described by the biharmonic equation
∇^4 ψ ≡ ∇^2 (∇^2 ψ) = 0.
By looking for separable solutions of the form ψ(r, θ) = rm^ cos 2θ, where (r, θ) are plane polar coordinates, find a solution inside the circle r = 1 that satisfies ψ = 0 and ∂ψ∂r = cos 2θ on r = 1 and ψ = 0 at r = 0. (^) [13]
Find also a solution outside the circle r = 1 that satisfies the same boundary conditions on r = 1 and is such that ∂ψ∂r → 0 as r → ∞. (^) [7]
[In plane polar coordinates (r, θ),
∇^2 φ =
r
∂r
r
∂φ ∂r
r^2
∂^2 φ ∂θ^2
.]
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3A What is a branch point of a function f (z)? (^) [2]
Find the branch points of f (z) = (z^2 − 1)^1 /^2 and describe how to make f (z) single valued by making (a) one branch cut and (b) two branch cuts in the complex plane. (^) [4]
For the shortest possible cut find the value of f (z) in terms of x and y, where z = x + iy, on either side of the cut. (^) [4]
Using integration by parts or otherwise show that
u(z) =
C
ezζ (ζ^2 − 1)^1 /^2
dζ
satisfies zu′′^ + u′^ − zu = 0
when the closed contour C encloses both branch points of f (z). (^) [5]
By shrinking the contour on to the branch cut and taking care with integrating around the branch points find u(0). (^) [5]
4A In terms of a Laurent expansion for a function f (z), in the complex plane, about z = z 0 define a pole, its order and its residue. (^) [4]
Find the poles, and their orders and residues, of
f (z) =
(z^2 + 1)^2 z[a^2 (z^2 + 1)^2 − (z^2 − 1)^2 ]
when 0 < a ≤ 1. Pay particular attention to the special case a = 1. (^) [9]
By integrating f (z) around the unit circle and applying the calculus of residues show that
I =
∫ (^2) π
0
a^2 + tan^2 θ
dθ =
2 π a(1 + a)
[7]
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5A Define the Laplace transform F (p) of a function f (t) and write down the Bromwich inversion formula for f (t) in terms of F (p) clearly specifying the path of integration. (^) [4]
Given a function h(λ), define the integral transform
H(s) =
0
λs−^1 h(λ) dλ.
By considering λ = e−t^ show that the inverse of this integral transform is
h(λ) =
2 πi
∫ (^) γ+i∞
γ−i∞
λ−sH(s) ds.
for a suitable γ. (^) [6]
Use this formula to determine h(λ) for 0 < λ < 1 when
(a) H(s) =
2 s − 1
and
(b) H(s) =
2 s+1(s + 1)
[10]
6C (a) Write down the transformation law for a rank n tensor. Use this to define an isotropic tensor. (^) [4]
(b) Show that any second rank tensor Pij can be written as a sum of a symmetric tensor and an antisymmetric tensor. Hence, or otherwise, show that Pij can be decomposed into the following terms Pij = P δij + Sij + ijkAk , (†)
where Sij is symmetric and traceless, Sii = 0. Give the tensors P , Sij and Ak explicitly in terms of Pij. (^) [8]
(c) The stress-strain equation, Pij = cijkpk ,
relates the stress Pij on a material to the resulting strain pij (both second rank tensors) through the elasticity tensor cijk` (fourth rank). For an isotropic material, use the stress- strain equation to express the decomposed components of Pij (i.e. P , Sij and Ak from (†)) in terms of the decomposed components of pij (denoted as p, sij , and ak). Hence, show that if the strain pij is symmetric then so is the stress Pij. (^) [8]
[You may assume that the most general fourth rank isotropic tensor is
cijk= λδij δk + μδikδj+ νδiδjk ,
where λ, μ and ν are scalars. The summation convention is assumed throughout.]
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9B For an arbitrary group G define what is meant by a conjugacy class of G. (^) [4]
Prove that no element of G can be in two different conjugacy classes and therefore that each element belongs to a unique conjugacy class. (^) [4]
If G is Abelian show that each element of G forms a class by itself. (^) [4]
The centre Z of the group G is defined as the set of elements of G which form a conjugacy class by themselves. Prove that Z is an Abelian subgroup of G (^) [6]
Find the centre of D 4 , the symmetry group of the square. (^) [2]
10B Define the terms representation and irreducible representation of a group G. (^) [3]
Consider the set Σ 3 of all possible permutations of three objects (a, b, c) with an operation defined as the successive application of the permutations. A 3 × 3 matrix representation for each element of the group is given by the following set of matrices,
I =
, A =
, B =
C =
, D =
, E =
Construct the multiplication table for this group based on the corresponding matrix multiplication or otherwise. What are the conjugacy classes? (^) [5]
From the knowledge of the order of the group, show that this is not an irreducible representation of Σ 3 and find the dimensions of all the irreducible representations of Σ 3. (^) [5]
Compute the character of each conjugacy class in the 3 × 3 matrix representation defined above. Then construct the table of characters for the irreducible representations, without constructing the representations explicitly. Decompose the 3 × 3 matrix represen- tation above in terms of the irreducible representations of the group of dimension 2 and
- [7]
∑ [The orthogonality relation for characters may be used as well as the relation α n
2 α =^ |G|^ where^ nα^ are the dimensions of the irreducible representations.]
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