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MATHEMATICAL TRIPOS Part IA
Tuesday 5th June 2007 1.30 pm to 4.30 pm
PAPER 3
Before you begin read these instructions carefully.
The examination paper is divided into two sections. Each question in Section II carries twice the number of marks of each question in Section I. Candidates may attempt all four questions from Section I and at most five questions from Section II. In Section II, no more than three questions on each course may be attempted.
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, marked A and D according to the code letter affixed to each question. Include in the same bundle all questions from Section I and II with the same code letter.
Attach a gold cover sheet to each bundle; write the code 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
1D Algebra and Geometry
Prove that every permutation of { 1 ,... , n} may be expressed as a product of disjoint cycles.
Let σ = (1234) and let τ = (345)(678). Write στ as a product of disjoint cycles. What is the order of στ?
2D Algebra and Geometry What does it mean to say that groups G and H are isomorphic?
Prove that no two of C 8 , C 4 × C 2 and C 2 × C 2 × C 2 are isomorphic. [Here Cn denotes the cyclic group of order n.]
Give, with justification, a group of order 8 that is not isomorphic to any of those three groups.
3A Vector Calculus (i) Give definitions for the unit tangent vector Tˆ and the curvature κ of a parametrised curve x(t) in R^3. Calculate Tˆ and κ for the circular helix
x(t) = (a cos t , a sin t , bt) ,
where a and b are constants.
(ii) Find the normal vector and the equation of the tangent plane to the surface S in R^3 given by z = x^2 y^3 − y + 1
at the point x = 1, y = 1, z = 1.
4A Vector Calculus
By using suffix notation, prove the following identities for the vector fields A and B in R^3 : ∇ · (A × B) = B · (∇ × A) − A · (∇ × B) ;
∇ × (A × B) = (B · ∇)A − B(∇ · A) − (A · ∇)B + A(∇ · B).
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7D Algebra and Geometry
Let A be a real symmetric n × n matrix. Prove that every eigenvalue of A is real, and that eigenvectors corresponding to distinct eigenvalues are orthogonal. Indicate clearly where in your argument you have used the fact that A is real.
What does it mean to say that a real n × n matrix P is orthogonal? Show that if P is orthogonal and A is as above then P −^1 AP is symmetric. If P is any real invertible matrix, must P −^1 AP be symmetric? Justify your answer.
Give, with justification, real 2×2 matrices B, C, D, E with the following properties: (i) B has no real eigenvalues;
(ii) C is not diagonalisable over C;
(iii) D is diagonalisable over C, but not over R; (iv) E is diagonalisable over R, but does not have an orthonormal basis of eigenvectors.
8D Algebra and Geometry
In the group of M¨obius maps, what is the order of the M¨obius map z 7 →
z
? What
is the order of the M¨obius map z 7 →
1 − z
Prove that every M¨obius map is conjugate either to a map of the form z 7 → μz (some μ ∈ C) or to the map z 7 → z + 1. Is z 7 → z + 1 conjugate to a map of the form z 7 → μz?
Let f be a M¨obius map of order n, for some positive integer n. Under the action on C ∪ {∞} of the group generated by f , what are the various sizes of the orbits? Justify your answer.
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9A Vector Calculus
(i) Define what is meant by a conservative vector field. Given a vector field A = (A 1 (x, y), A 2 (x, y)) and a function ψ(x, y) defined in R^2 , show that, if ψA is a conservative vector field, then
ψ
∂A 1
∂y
∂A 2
∂x
= A 2
∂ψ ∂x
− A 1
∂ψ ∂y
(ii) Given two functions P (x, y) and Q(x, y) defined in R^2 , prove Green’s theorem, ∮
C
(P dx + Q dy) =
R
∂Q
∂x
∂P
∂y
dx dy ,
where C is a simple closed curve bounding a region R in R^2.
Through an appropriate choice for P and Q, find an expression for the area of the region R, and apply this to evaluate the area of the ellipse bounded by the curve
x = a cos θ , y = b sin θ , 0 ≤ θ ≤ 2 π.
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11A Vector Calculus
The function φ(x, y, z) satisfies ∇^2 φ = 0 in V and φ = 0 on S, where V is a region of R^3 which is bounded by the surface S. Prove that φ = 0 everywhere in V.
Deduce that there is at most one function ψ(x, y, z) satisfying ∇^2 ψ = ρ in V and ψ = f on S, where ρ(x, y, z) and f (x, y, z) are given functions.
Given that the function ψ = ψ(r) depends only on the radial coordinate r = |x|, use Cartesian coordinates to show that
∇ψ =
r
dψ dr
x , ∇^2 ψ =
r
d^2 (rψ) dr^2
Find the general solution in this radial case for ∇^2 ψ = c where c is a constant.
Find solutions ψ(r) for a solid sphere of radius r = 2 with a central cavity of radius r = 1 in the following three regions:
(i) 0 6 r 6 1 where ∇^2 ψ = 0 and ψ(1) = 1 and ψ bounded as r → 0;
(ii) 1 6 r 6 2 where ∇^2 ψ = 1 and ψ(1) = ψ(2) = 1; (iii) r > 2 where ∇^2 ψ = 0 and ψ(2) = 1 and ψ → 0 as r → ∞.
12A Vector Calculus
Show that any second rank Cartesian tensor Pij in R^3 can be written as a sum of a symmetric tensor and an antisymmetric tensor. Further, show that Pij can be decomposed into the following terms Pij = P δij + Sij + ijkAk , (†)
where Sij is symmetric and traceless. Give expressions for P , Sij and Ak explicitly in terms of Pij.
For an isotropic material, the stress Pij can be related to the strain Tij through the stress–strain relation, Pij = cijkl Tkl , where the elasticity tensor is given by
cijkl = αδij δkl + βδikδjl + γδilδjk
and α, β and γ are scalars. As in (†), the strain Tij can be decomposed into its trace T , a symmetric traceless tensor Wij and a vector Vk. Use the stress–strain relation to express each of T , Wij and Vk in terms of P , Sij and Ak.
Hence, or otherwise, show that if Tij is symmetric then so is Pij. Show also that the stress-strain relation can be written in the form
Pij = λ δij Tkk + μ Tij ,
where μ and λ are scalars.
END OF PAPER
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