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MATHEMATICAL TRIPOS Part IB
Wednesday 6 June 2007 1.30 to 4.
PAPER 2
Before you begin read these instructions carefully.
Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt at most four questions from Section I and at most six questions from Section II.
Complete answers are preferred to fragments.
Write on one side of the paper only and begin each answer on a separate sheet.
Write legibly; otherwise, you place yourself at a grave disadvantage.
At the end of the examination:
Tie up your answers in separate bundles labelled A, B,... , H according to the examiner letter affixed to each question, including in the same bundle questions from Sections I and II with the same examiner letter.
Attach a completed gold cover sheet to each bundle; write the examiner letter in the box marked ‘Examiner Letter’ on the cover sheet.
You must also complete a green master cover sheet listing all the questions you have attempted.
Every cover sheet must bear your examination number and desk number.
STATIONERY REQUIREMENTS SPECIAL REQUIREMENTS Gold cover sheet None Green master cover sheet
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
SECTION I
1G Linear Algebra
Suppose that S, T are endomorphisms of the 3-dimensional complex vector space C^3 and that the eigenvalues of each of them are 1, 2, 3. What are their characteristic and minimal polynomials? Are they conjugate?
2G Groups, Rings and Modules
Define the term Euclidean domain. Show that the ring of integers Z is a Euclidean domain.
3H Analysis II
For integers a and b, define d(a, b) to be 0 if a = b, or (^21) n if a 6 = b and n is the largest non-negative integer such that a − b is a multiple of 2n. Show that d is a metric on the integers Z.
Does the sequence xn = 2n^ − 1 converge in this metric?
4A Metric and Topological Spaces Are the following statements true or false? Give a proof or a counterexample as appropriate.
(i) If f : X → Y is a continuous map of topological spaces and S ⊆ X is compact then f (S) is compact.
(ii) If f : X → Y is a continuous map of topological spaces and K ⊆ Y is compact then f −^1 (K) = {x ∈ X : f (x) ∈ K}} is compact.
(iii) If a metric space M is complete and a metric space T is homeomorphic to M then T is complete.
Paper 2
8D Fluid Dynamics
An incompressible, inviscid fluid occupies the region beneath the free surface y = η(x, t) and moves with a velocity field given by the velocity potential φ(x, y, t); gravity acts in the −y direction. Derive the kinematic and dynamic boundary conditions that must be satisfied by φ on y = η(x, t).
[You may assume Bernoulli’s integral of the equation of motion:
p ρ
∂φ ∂t
|∇φ|^2 + gy = F (t). ]
In the absence of waves, the fluid has uniform velocity U in the x direction. Derive the linearised form of the above boundary conditions for small amplitude waves, and verify that they and Laplace’s equation are satisfied by the velocity potential
φ = U x + Re{beky^ ei(kx−ωt)} ,
where |kb| U , with a corresponding expression for η, as long as
(ω − kU )^2 = gk.
What are the propagation speeds of waves with a given wave-number k?
9C Optimization
Consider the game with payoff matrix
where the (i, j) entry is the payoff to the row player if the row player chooses row i and the column player chooses column j.
Find the value of the game and the optimal strategies for each player.
Paper 2
SECTION II
10G Linear Algebra
Suppose that P is the complex vector space of complex polynomials in one variable, z.
(i) Show that the form 〈 , 〉 defined by
〈f, g〉 =
2 π
∫ (^2) π
0
f
eiθ^
· g (eiθ^ ) dθ
is a positive definite Hermitian form on P.
(ii) Find an orthonormal basis of P for this form, in terms of the powers of z. (iii) Generalize this construction to complex vector spaces of complex polynomials in any finite number of variables.
11G Groups, Rings and Modules (i) Give an example of a Noetherian ring and of a ring that is not Noetherian. Justify your answers.
(ii) State and prove Hilbert’s basis theorem.
12A Geometry
(i) The spherical circle with centre P ∈ S^2 and radius r, 0 < r < π, is the set of all points on the unit sphere S^2 at spherical distance r from P. Find the circumference of a spherical circle with spherical radius r. Compare, for small r, with the formula for a Euclidean circle and comment on the result.
(ii) The cross ratio of four distinct points zi in C is
(z 4 − z 1 )(z 2 − z 3 ) (z 4 − z 3 )(z 2 − z 1 )
Show that the cross-ratio is a real number if and only if z 1 , z 2 , z 3 , z 4 lie on a circle or a line.
[You may assume that M¨obius transformations preserve the cross-ratio.]
Paper 2 [TURN OVER
15D Methods
Let y 0 (x) be a non-zero solution of the Sturm-Liouville equation
L(y 0 ; λ 0 ) ≡
d dx
p(x)
dy 0 dx
- (q(x) + λ 0 w(x)) y 0 = 0
with boundary conditions y 0 (0) = y 0 (1) = 0. Show that, if y(x) and f (x) are related by
L(y; λ 0 ) = f ,
with y(x) satisfying the same boundary conditions as y 0 (x), then
∫ (^1)
0
y 0 f dx = 0. (∗)
Suppose that y 0 is normalised so that ∫ (^1)
0
wy 02 dx = 1 ,
and consider the problem
L(y; λ) = y^3 ; y(0) = y(1) = 0.
By choosing f appropriately in (∗) deduce that, if
λ − λ 0 = ^2 μ [μ = O(1), 1 ] , and y(x) = y 0 (x) + ^2 y 1 (x)
then
μ =
0
y 04 dx + O().
Paper 2 [TURN OVER
16B Quantum Mechanics
Write down the angular momentum operators L 1 , L 2 , L 3 in terms of the position and momentum operators, x and p, and the commutation relations satisfied by x and p.
Verify the commutation relations
[Li, Lj ] = iℏijkLk.
Further, show that [Li, pj ] = iℏijkpk.
A wave-function Ψ 0 (r) is spherically symmetric. Verify that
LΨ 0 (r) = 0.
Consider the vector function Φ = ∇Ψ 0 (r). Show that Φ 3 and Φ 1 ± iΦ 2 are eigenfunctions of L 3 with eigenvalues 0, ±ℏ respectively.
17E Electromagnetism
If S is a fixed surface enclosing a volume V , use Maxwell’s equations to show that
d dt
V
0 E^2 +
2 μ 0
B^2
dV +
S
P · n dS = −
V
j · E dV ,
where P = (E × B)/μ 0. Give a physical interpretation of each term in this equation.
Show that Maxwell’s equations for a vacuum permit plane wave solutions with E = E 0 (0, 1 , 0)cos(kx − ωt) with E 0 , k and ω constants, and determine the relationship between k and ω.
Find also the corresponding B(x, t) and hence the time average < P >. What does < P > represent in this case?
18F Numerical Analysis
For a symmetric, positive definite matrix A with the spectral radius ρ(A), the linear system Ax = b is solved by the iterative procedure
x(k+1)^ = x(k)^ − τ (Ax(k)^ − b), k ≥ 0 ,
where τ is a real parameter. Find the range of τ that guarantees convergence of x(k)^ to the exact solution for any choice of x(0).
Paper 2
