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MATHEMATICAL TRIPOS Part IB
Wednesday 4 June 2003 1.30 to 4.
PAPER 2
Before you begin read these instructions carefully.
Each question in Section II carries twice the credit of each question in Section I. You should 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 code letter affixed to each question, including in the same bundle questions from Sections I and II with the same code letter.
Attach a completed blue cover sheet to each bundle; write the code letter in the box marked ‘SECTION’ 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.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
SECTION I
1F Analysis II
Explain what it means for a function f : R^2 → R^1 to be differentiable at a point (a, b). Show that if the partial derivatives ∂f /∂x and ∂f /∂y exist in a neighbourhood of (a, b) and are continuous at (a, b) then f is differentiable at (a, b).
Let f (x, y) =
xy x^2 + y^2
((x, y) 6 = (0, 0))
and f (0, 0) = 0. Do the partial derivatives of f exist at (0, 0)? Is f differentiable at (0, 0)? Justify your answers.
2C Methods
Explain briefly why the second-rank tensor ∫
S
xixj dS(x)
is isotropic, where S is the surface of the unit sphere centred on the origin.
A second-rank tensor is defined by
Tij (y) =
S
(yi − xi)(yj − xj ) dS(x) ,
where S is the surface of the unit sphere centred on the origin. Calculate T (y) in the form
Tij = λδij + μyiyj ,
where λ and μ are to be determined.
By considering the action of T on y and on vectors perpendicular to y, determine the eigenvalues and associated eigenvectors of T.
3H Statistics Let X 1 ,... , Xn be a random sample from the N (θ, σ^2 ) distribution, and suppose that the prior distribution for θ is N (μ, τ 2 ), where σ^2 , μ, τ 2 are known. Determine the posterior distribution for θ, given X 1 ,... , Xn, and the best point estimate of θ under both quadratic and absolute error loss.
Paper 2
8G Quadratic Mathematics
Let U be a finite-dimensional real vector space and b a positive definite symmetric bilinear form on U ×U. Let ψ : U → U be a linear map such that b(ψ(x), y)+b(x, ψ(y)) = 0 for all x and y in U. Prove that if ψ is invertible, then the dimension of U must be even. By considering the restriction of ψ to its image or otherwise, prove that the rank of ψ is always even.
9A Quantum Mechanics
What is meant by the statement than an operator is hermitian?
A particle of mass m moves in the real potential V (x) in one dimension. Show that the Hamiltonian of the system is hermitian.
Show that d dt
〈x〉 =
m
〈p〉 ,
d dt
〈p〉 = 〈−V ′(x)〉 ,
where p is the momentum operator and 〈A〉 denotes the expectation value of the operator A.
Paper 2
SECTION II
10F Analysis II
Let V be the space of n × n real matrices. Show that the function
N (A) = sup {‖Ax‖ : x ∈ Rn, ‖x‖ = 1}
(where ‖ − ‖ denotes the usual Euclidean norm on Rn) defines a norm on V. Show also that this norm satisfies N (AB) 6 N (A)N (B) for all A and B, and that if N (A) < then all entries of A have absolute value less than . Deduce that any function f : V → R such that f (A) is a polynomial in the entries of A is continuously differentiable.
Now let d : V → R be the mapping sending a matrix to its determinant. By considering d(I + H) as a polynomial in the entries of H, show that the derivative d′(I) is the function H 7 → tr H. Deduce that, for any A, d′(A) is the mapping H 7 → tr((adj A)H), where adj A is the adjugate of A, i.e. the matrix of its cofactors.
[Hint: consider first the case when A is invertible. You may assume the results that the set U of invertible matrices is open in V and that its closure is the whole of V , and the identity (adj A)A = det A.I.]
11C Methods
State the transformation law for an nth-rank tensor Tij···k. Show that the fourth-rank tensor
cijkl = α δij δkl + β δik δjl + γ δil δjk
is isotropic for arbitrary scalars α, β and γ.
The stress σij and strain eij in a linear elastic medium are related by
σij = cijkl ekl.
Given that eij is symmetric and that the medium is isotropic, show that the stress-strain relationship can be written in the form
σij = λ ekk δij + 2μ eij.
Show that eij can be written in the form eij = pδij + dij , where dij is a traceless tensor and p is a scalar to be determined. Show also that necessary and sufficient conditions for the stored elastic energy density E = 12 σij eij to be non-negative for any deformation of the solid are that μ ≥ 0 and λ ≥ − 23 μ.
Paper 2 [TURN OVER
15E Linear Mathematics
(a) Let A = (aij ) be an m × n matrix and for each k 6 n let Ak be the m × k matrix formed by the first k columns of A. Suppose that n > m. Explain why the nullity of A is non-zero. Prove that if k is minimal such that Ak has non-zero nullity, then the nullity of Ak is 1.
(b) Suppose that no column of A consists entirely of zeros. Deduce from (a) that there exist scalars b 1 ,... , bk (where k is defined as in (a)) such that
∑k j=1 aij^ bj^ = 0 for every i 6 m, but whenever λ 1 ,... , λk are distinct real numbers there is some i 6 m such that
∑k j=1 aij^ λj^ bj^6 = 0. (c) Now let v 1 , v 2 ,... , vm and w 1 , w 2 ,... , wm be bases for the same real m- dimensional vector space. Let λ 1 , λ 2 ,... , λn be distinct real numbers such that for every j the vectors v 1 + λj w 1 ,... , vm + λj wm are linearly dependent. For each j, let a 1 j ,... , amj be scalars, not all zero, such that
∑m i=1 aij^ (vi^ +^ λj^ wi) =^0. By applying the result of (b) to the matrix (aij ), deduce that n 6 m.
(d) It follows that the vectors v 1 +λw 1 ,... , vm +λwm are linearly dependent for at most m values of λ. Explain briefly how this result can also be proved using determinants.
16B Complex Methods
(a) Show that if f satisfies the equation
f ′′(x) − x^2 f (x) = μf (x), x ∈ R, (∗)
where μ is a constant, then its Fourier transform f̂ satisfies the same equation, i.e.
f̂ ′′(λ) − λ^2 f̂ (λ) = μf̂ (λ).
(b) Prove that, for each n ≥ 0, there is a polynomial pn(x) of degree n, unique up to multiplication by a constant, such that
fn(x) = pn(x)e−x
(^2) / 2
is a solution of (∗) for some μ = μn.
(c) Using the fact that g(x) = e−x
(^2) / 2 satisfies ̂g = cg for some constant c, show that the Fourier transform of fn has the form
̂ fn(λ) = qn(λ)e−λ^2 /^2
where qn is also a polynomial of degree n.
(d) Deduce that the fn are eigenfunctions of the Fourier transform operator, i.e. ̂ fn(x) = cnfn(x) for some constants cn.
Paper 2 [TURN OVER
17G Quadratic Mathematics
Let S be the set of all 2 × 2 complex matrices A which are hermitian, that is,
A∗^ = A, where A∗^ = A
t . (a) Show that S is a real 4-dimensional vector space. Consider the real symmetric bilinear form b on this space defined by
b(A, B) = 12 (tr(A B) − tr(A) tr(B)).
Prove that b(A, A) = −det A and b(A, I) = − 12 tr(A), where I denotes the identity matrix.
(b) Consider the three matrices
A 1 =
, A 2 =
and A 3 =
0 −i i 0
Prove that the basis I, A 1 , A 2 , A 3 of S diagonalizes b. Hence or otherwise find the rank and signature of b.
(c) Let Q be the set of all 2 × 2 complex matrices C which satisfy C + C∗^ = tr(C) I. Show that Q is a real 4-dimensional vector space. Given C ∈ Q, put
Φ(C) =
1 − i 2
tr(C) I + i C.
Show that Φ takes values in S and is a linear isomorphism between Q and S.
(d) Define a real symmetric bilinear form on Q by setting c(C, D) = − 12 tr(C D), C, D ∈ Q. Show that b(Φ(C), Φ(D)) = c(C, D) for all C, D ∈ Q. Find the rank and signature of the symmetric bilinear form c defined on Q.
18A Quantum Mechanics
A particle of mass m and energy E moving in one dimension is incident from the left on a potential barrier V (x) given by
V (x) =
{ V
0 0 6 x^6 a 0 otherwise
with V 0 > 0.
In the limit V 0 → ∞, a → 0 with V 0 a = U held fixed, show that the transmission probability is
T =
mU 2 2 Eℏ^2
Paper 2
