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MATHEMATICAL TRIPOS Part IB
Thursday 5 June 2008 9.00 to 12.
PAPER 3
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 REQUIRMENTS 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 Groups, Rings and Modules
Let G be the abelian group generated by elements a, b, c, d subject to the relations
4 a − 2 b + 2c + 12d = 0, − 2 b + 2c = 0, 2 b + 2c = 0, 8 a + 4c + 24d = 0.
Express G as a product of cyclic groups, and find the number of elements of G of order 2.
2G Geometry
A smooth surface in R^3 has parametrization
σ(u, v) =
u −
u^3 3
v^3 3
Show that a unit normal vector at the point σ(u, v) is
( − 2 u 1 + u^2 + v^2
2 v 1 + u^2 + v^2
1 − u^2 − v^2 1 + u^2 + v^2
and that the curvature is
(1 + u^2 + v^2 )^4
3F Analysis II Explain what it means for a function f (x, y) of two variables to be differentiable at a point (x 0 , y 0 ). If f is differentiable at (x 0 , y 0 ), show that for any α the function gα defined by gα(t) = f (x 0 + t cos α, y 0 + t sin α)
is differentiable at t = 0, and find its derivative in terms of the partial derivatives of f at (x 0 , y 0 ).
Consider the function f defined by
f (x, y) = (x^2 y + xy^2 )/(x^2 + y^2 ) ((x, y) 6 = (0, 0)) = 0 ((x, y) = (0, 0)).
Is f differentiable at (0, 0)? Justify your answer.
Paper 3
7A Quantum Mechanics
Write down a formula for the orbital angular momentum operator Lˆ. Show that its components satisfy [Li, Lj ] = iℏ ijk Lk.
If L 3 ψ = 0, show that (L 1 ± iL 2 )ψ are also eigenvectors of L 3 , and find their eigenvalues.
8H Statistics If X 1 ,... , Xn is a sample from a density f (·|θ) with θ unknown, what is a 95% confidence set for θ?
In the case where the Xi are independent N (μ, σ^2 ) random variables with σ^2 known, μ unknown, find (in terms of σ^2 ) how large the size n of the sample must be in order for there to exist a 95% confidence interval for μ of length no more than some given ε > 0.
[Hint: If Z ∼ N (0, 1) then P (Z > 1 .960) = 0. 025 .]
9H Markov Chains
What does it mean to say that a Markov chain is recurrent?
Stating clearly any general results to which you appeal, prove that the symmetric simple random walk on Z is recurrent.
Paper 3
SECTION II
10E Linear Algebra
Let k = R or C. What is meant by a quadratic form q : kn^ → k? Show that there is a basis {v 1 ,... , vn} for kn^ such that, writing x = x 1 v 1 +... + xnvn, we have q(x) = a 1 x^21 +... + anx^2 n for some scalars a 1 ,... , an ∈ {− 1 , 0 , 1 }.
Suppose that k = R. Define the rank and signature of q and compute these quantities for the form q : R^3 → R given by q(x) = − 3 x^21 + x^22 + 2x 1 x 2 − 2 x 1 x 3 + 2x 2 x 3.
Suppose now that k = C and that q 1 ,... , qd : Cn^ → C are quadratic forms. If n > 2 d, show that there is some nonzero x ∈ Cn^ such that q 1 (x) =... = qd(x) = 0.
11G Groups, Rings and Modules
What is a Euclidean domain? Show that a Euclidean domain is a principal ideal domain.
Show that Z[
−7] is not a Euclidean domain (for any choice of norm), but that the ring
Z
[ 1 + √− 7
]
is Euclidean for the norm function N (z) = z z¯.
Paper 3 [TURN OVER
14E Complex Analysis
State and prove Rouch´e’s theorem, and use it to count the number of zeros of 3 z^9 + 8z^6 + z^5 + 2z^3 + 1 inside the annulus {z : 1 < |z| < 2 }.
Let (pn)∞ n=1 be a sequence of polynomials of degree at most d with the property that pn(z) converges uniformly on compact subsets of C as n → ∞. Prove that there is a polynomial p of degree at most d such that pn → p uniformly on compact subsets of C. [If you use any results about uniform convergence of analytic functions, you should prove them.]
Suppose that p has d distinct roots z 1 ,... , zd. Using Rouch´e’s theorem, or otherwise, show that for each i there is a sequence (zi,n)∞ n=1 such that pn(zi,n) = 0 and zi,n → zi as n → ∞.
Paper 3 [TURN OVER
15D Methods
Let λ 1 < λ 2 <... λn... and y 1 (x), y 2 (x),... yn(x)... be the eigenvalues and corresponding eigenfunctions for the Sturm–Liouville system
Lyn = λnw(x)yn,
where
Ly ≡
d dx
−p(x)
dy dx
with p(x) > 0 and w(x) > 0. The boundary conditions on y are that y(0) = y(1) = 0.
Show that two distinct eigenfunctions are orthogonal in the sense that ∫ (^1)
0
wynym dx = δnm
0
wy n^2 dx.
Show also that if y has the form
y =
∑^ ∞
n=
anyn,
with an being independent of x, then
∫ (^1) ∫^0 yLy dx 1 0 wy
(^2) dx
≥ λ 1.
Assuming that the eigenfunctions are complete, deduce that a solution of the diffusion equation, ∂y ∂t
w
Ly,
that satisfies the boundary conditions given above is such that
1 2
d dt
0
wy^2 dx
≤ −λ 1
0
wy^2 dx.
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17B Electromagnetism
(i) From Maxwell’s equations in vacuum,
∇ · E = 0 ∇ × E = −
∂B
∂t
∇ · B = 0 ∇ × B = μ 0 0
∂E
∂t
obtain the wave equation for the electric field E. [You may find the following identity useful: ∇ × (∇ × A) = ∇(∇ · A) − ∇^2 A.]
(ii) If the electric and magnetic fields of a monochromatic plane wave in vacuum are E(z, t) = E 0 ei(kz−ωt)^ and B(z, t) = B 0 ei(kz−ωt)^ ,
show that the corresponding electromagnetic waves are transverse (that is, both fields have no component in the direction of propagation).
(iii) Use Faraday’s law for these fields to show that
B 0 =
k ω
(ˆez × E 0 ).
(iv) Explain with symmetry arguments how these results generalise to
E(r, t) = E 0 ei(k·r−ωt)^ nˆ and B(r, t) =
c
E 0 ei(k·r−ωt)(ˆk × ˆn) ,
where ˆn is the polarisation vector, i.e., the unit vector perpendicular to the direction of motion and along the direction of the electric field, and ˆk is the unit vector in the direction of propagation of the wave.
(v) Using Maxwell’s equations in vacuum prove that: ∮
A
(1/μ 0 )(E × B) · dA = −
∂t
V
0 E^2
B^2
2 μ 0
dV , (1)
where V is the closed volume and A is the bounding surface. Comment on the differing time dependencies of the left-hand-side of (1) for the case of (a) linearly-polarized and (b) circularly-polarized monochromatic plane waves.
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18B Fluid Dynamics
An ideal liquid contained within a closed circular cylinder of radius a rotates about the axis of the cylinder (assume this axis to be in the vertical z-direction).
(i) Prove that the equation of continuity and the boundary conditions are satisfied by the velocity v = Ω × r, where Ω = Ωˆez is the angular velocity, with ˆez the unit vector in the z-direction, which depends only on time, and r is the position vector measured from a point on the axis of rotation.
(ii) Calculate the angular momentum M = ρ
(r × v)dV per unit length of the cylinder.
(iii) Suppose the the liquid starts from rest and flows under the action of an external force per unit mass f = (αx + βy, γx + δy, 0). By taking the curl of the Euler equation, prove that dΩ dt
(γ − β).
(iv) Find the pressure.
19D Numerical Analysis Starting from the Taylor formula for f (x) ∈ Ck+1[a, b] with an integral remainder term, show that the error of an approximant L(f ) can be written in the form (Peano kernel theorem)
L(f ) =
k!
∫ (^) b
a
K(θ)f (k+1)(θ)dθ,
when L(f ), which is identically zero if f (x) is a polynomial of degree k, satisfies conditions that you should specify. Give an expression for K(θ).
Hence determine the minimum value of c in the inequality |L(f )| ≤ c‖f ′′′‖∞ ,
when
L(f ) = f ′(1) −
(f (2) − f (0)) for f (x) ∈ C^3 [0, 2].
20H Optimization
Use the simplex algorithm to solve the problem max x 1 + 2x 2 − 6 x 3
subject to x 1 , x 2 > 0, |x 3 | 6 5, and
x 1 + x 2 + x 3 6 7 , 2 x 2 + x 3 > 1.
END OF PAPER
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