Download Mathematical Tripos Part IA Paper 3 Examination Document and more Exams Mathematics in PDF only on Docsity!
MATHEMATICAL TRIPOS Part IA
Tuesday, 5 June, 2012 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, B, C, D, E and F 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 completed gold cover sheet to each bundle.
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 sheets 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
1E Groups State Lagrange’s Theorem. Deduce that if G is a finite group of order n, then the order of every element of G is a divisor of n.
Let G be a group such that, for every g ∈ G, g^2 = e. Show that G is abelian. Give an example of a non-abelian group in which every element g satisfies g^4 = e.
2E Groups What is a cycle in the symmetric group Sn? Show that a cycle of length p and a cycle of length q in Sn are conjugate if and only if p = q.
Suppose that p is odd. Show that any two p-cycles in Ap+2 are conjugate. Are any two 3-cycles in A 4 conjugate? Justify your answer.
3C Vector Calculus Define what it means for a differential P dx+Q dy to be exact, and derive a necessary condition on P (x, y) and Q(x, y) for this to hold. Show that one of the following two differentials is exact and the other is not:
y^2 dx + 2xy dy , y^2 dx + xy^2 dy.
Show that the differential which is not exact can be written in the form g df for functions f (x, y) and g(y), to be determined.
4C Vector Calculus What does it mean for a second-rank tensor Tij to be isotropic? Show that δij is isotropic. By considering rotations through π/2 about the coordinate axes, or otherwise, show that the most general isotropic second-rank tensor in R^3 has the form Tij = λδij , for some scalar λ.
Part IA, Paper 3
7E Groups Let Fp be the set of (residue classes of) integers mod p, and let
G =
a b c d
: a, b, c, d ∈ Fp, ad − bc 6 = 0
Show that G is a group under multiplication. [You may assume throughout this question that multiplication of matrices is associative.]
Let X be the set of 2-dimensional column vectors with entries in Fp. Show that the mapping G × X → X given by (( a b c d
x y
ax + by cx + dy
is a group action.
Let g ∈ G be an element of order p. Use the orbit-stabilizer theorem to show that there exist x, y ∈ Fp, not both zero, with
g
x y
x y
Deduce that g is conjugate in G to the matrix ( 1 1 0 1
Part IA, Paper 3
8E Groups Let p be a prime number, and a an integer with 1 6 a 6 p − 1. Let G be the Cartesian product
G = { (x, u) | x ∈ { 0 , 1 ,... , p − 2 }, u ∈ { 0 , 1 ,... , p − 1 } }
Show that the binary operation
(x, u) ∗ (y, v) = (z, w)
where
z ≡ x + y (mod p − 1) w ≡ ayu + v (mod p)
makes G into a group. Show that G is abelian if and only if a = 1. Let H and K be the subsets
H = { (x, 0) | x ∈ { 0 , 1 ,... , p − 2 } }, K = { (0, u) | u ∈ { 0 , 1 ,... , p − 1 } }
of G. Show that K is a normal subgroup of G, and that H is a subgroup which is normal if and only if a = 1. Find a homomorphism from G to another group whose kernel is K.
9C Vector Calculus State Stokes’ Theorem for a vector field B(x) on R^3. Consider the surface S defined by
z = x^2 + y^2 ,
6 z 6 1.
Sketch the surface and calculate the area element dS in terms of suitable coordinates or parameters. For the vector field
B = (−y^3 , x^3 , z^3 )
compute ∇ × B and calculate I =
S (∇ ×^ B)^ ·^ dS. Use Stokes’ Theorem to express I as an integral over ∂S and verify that this gives the same result.
Part IA, Paper 3 [TURN OVER
12C Vector Calculus (i) Let V be a bounded region in R^3 with smooth boundary S = ∂V. Show that Poisson’s equation in V ∇^2 u = ρ has at most one solution satisfying u = f on S, where ρ and f are given functions. Consider the alternative boundary condition ∂u/∂n = g on S, for some given function g, where n is the outward pointing normal on S. Derive a necessary condition in terms of ρ and g for a solution u of Poisson’s equation to exist. Is such a solution unique? (ii) Find the most general spherically symmetric function u(r) satisfying
∇^2 u = 1
in the region r = |r| 6 a for a > 0. Hence in each of the following cases find all possible solutions satisfying the given boundary condition at r = a:
(a) u = 0 ,
(b) ∂u ∂n = 0.
Compare these with your results in part (i).
END OF PAPER
Part IA, Paper 3
