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MATHEMATICAL TRIPOS Part IB
Thursday 5 June 2003 9 to 12
PAPER 3
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
Let V be the vector space of continuous real-valued functions on [− 1 , 1]. Show that the function
‖f ‖ =
− 1
|f (x)| dx
defines a norm on V.
Let fn(x) = xn. Show that (fn) is a Cauchy sequence in V. Is (fn) convergent? Justify your answer.
2D Methods Consider the path between two arbitrary points on a cone of interior angle 2α. Show that the arc-length of the path r(θ) is given by
∫ (r^2 + r′^2 cosec^2 α)^1 /^2 dθ ,
where r′^ = drdθ. By minimizing the total arc-length between the points, determine the equation for the shortest path connecting them.
3E Further Analysis (a) Let f : C → C be an analytic function such that |f (z)| 6 1 + |z|^1 /^2 for every z. Prove that f is constant.
(b) Let f : C → C be an analytic function such that Re (f (z)) > 0 for every z. Prove that f is constant.
4F Geometry Show that any isometry of Euclidean 3-space which fixes the origin can be written as a composite of at most three reflections in planes through the origin, and give (with justification) an example of an isometry for which three reflections are necessary.
Paper 3
8C Fluid Dynamics
Show that the velocity field
u = U +
Γ × r 2 πr^2
where U = (U, 0 , 0), Γ = (0, 0 , Γ) and r = (x, y, 0) in Cartesian coordinates (x, y, z), represents the combination of a uniform flow and the flow due to a line vortex. Define and evaluate the circulation of the vortex.
Show that (^) ∮
CR
(u · n)u dl → 12 Γ × U as R → ∞,
where CR is a circle x^2 + y^2 = R^2 , z = const. Explain how this result is related to the lift force on a two-dimensional aerofoil or other obstacle.
9G Quadratic Mathematics
Let f (x, y) = ax^2 + bxy + cy^2 be a binary quadratic form with integer coefficients. Explain what is meant by the discriminant d of f. State a necessary and sufficient condition for some form of discriminant d to represent an odd prime number p. Using this result or otherwise, find the primes p which can be represented by the form x^2 + 3y^2.
10A Special Relativity
What are the momentum and energy of a photon of wavelength λ?
A photon of wavelength λ is incident on an electron. After the collision, the photon has wavelength λ′. Show that
λ′^ − λ =
h mc
(1 − cos θ),
where θ is the scattering angle and m is the electron rest mass.
Paper 3
SECTION II
11F Analysis II
State and prove the Contraction Mapping Theorem. Let (X, d) be a bounded metric space, and let F denote the set of all continuous maps X → X. Let ρ : F × F → R be the function
ρ(f, g) = sup{d(f (x), g(x)) : x ∈ X}.
Show that ρ is a metric on F , and that (F, ρ) is complete if (X, d) is complete. [You may assume that a uniform limit of continuous functions is continuous.]
Now suppose that (X, d) is complete. Let C ⊆ F be the set of contraction mappings, and let θ : C → X be the function which sends a contraction mapping to its unique fixed point. Show that θ is continuous. [Hint: fix f ∈ C and consider d(θ(g), f (θ(g))), where g ∈ C is close to f .]
12D Methods The transverse displacement y(x, t) of a stretched string clamped at its ends x = 0, l satisfies the equation
∂^2 y ∂t^2
= c^2
∂^2 y ∂x^2
− 2 k
∂y ∂t
, y(x, 0) = 0,
∂y ∂t
(x, 0) = δ(x − a) ,
where c > 0 is the wave velocity, and k > 0 is the damping coefficient. The initial conditions correspond to a sharp blow at x = a at time t = 0.
(a) Show that the subsequent motion of the string is given by
y(x, t) =
α^2 n − k^2
n
2 e−kt^ sin
αna c
sin
αnx c
sin /(
α^2 n − k^2 t)
where αn = πcn/l.
(b) Describe what happens in the limits of small and large damping. What critical parameter separates the two cases?
Paper 3 [TURN OVER
16B Numerical Analysis
The functions H 0 , H 1 ,... are generated by the Rodrigues formula:
Hn(x) = (−1)nex
(^2) dn dxn^
e−x
2 .
(a) Show that Hn is a polynomial of degree n, and that the Hn are orthogonal with respect to the scalar product
(f, g) =
−∞
f (x)g(x)e−x
2 dx.
(b) By induction or otherwise, prove that the Hn satisfy the three-term recurrence relation Hn+1(x) = 2xHn(x) − 2 nHn− 1 (x).
[Hint: you may need to prove the equality H n′(x) = 2nHn− 1 (x) as well.]
17G Linear Mathematics Define the determinant det(A) of an n × n complex matrix A. Let A 1 ,... , An be the columns of A, let σ be a permutation of { 1 ,... , n} and let Aσ^ be the matrix whose columns are Aσ(1),... , Aσ(n). Prove from your definition of determinant that det(Aσ^ ) = (σ) det(A), where (σ) is the sign of the permutation σ. Prove also that det(A) = det(At).
Define the adjugate matrix adj(A) and prove from your definitions that A adj(A) = adj(A) A = det(A) I, where I is the identity matrix. Hence or otherwise, prove that if det(A) 6 = 0, then A is invertible.
Let C and D be real n × n matrices such that the complex matrix C + iD is invertible. By considering det(C + λ D) as a function of λ or otherwise, prove that there exists a real number λ such that C + λ D is invertible. [You may assume that if a matrix A is invertible, then det(A) 6 = 0.]
Deduce that if two real matrices A and B are such that there exists an invertible complex matrix P with P −^1 A P = B, then there exists an invertible real matrix Q such that Q−^1 A Q = B.
Paper 3 [TURN OVER
18C Fluid Dynamics
State the form of Bernoulli’s theorem appropriate for an unsteady irrotational motion of an inviscid incompressible fluid in the absence of gravity.
Water of density ρ is driven through a tube of length L and internal radius a by the pressure exerted by a spherical, water-filled balloon of radius R(t) attached to one end of the tube. The balloon maintains the pressure of the water entering the tube at 2γ/R in excess of atmospheric pressure, where γ is a constant. It may be assumed that the water exits the tube at atmospheric pressure. Show that
R^3 R¨ + 2R^2 R˙^2 = −
γa^2 2 ρL
Solve equation (†), by multiplying through by 2R R˙ or otherwise, to obtain
t = R^20
2 ρL γa^2
) 1 / 2 [
π 4
θ 2
sin 2θ
]
where θ = sin−^1 (R/R 0 ) and R 0 is the initial radius of the balloon. Hence find the time when R = 0.
19G Quadratic Mathematics
Let U be a finite-dimensional real vector space endowed with a positive definite inner product. A linear map τ : U → U is said to be an orthogonal projection if τ is self-adjoint and τ 2 = τ.
(a) Prove that for every orthogonal projection τ there is an orthogonal decomposi- tion U = ker(τ ) ⊕ im(τ ).
(b) Let φ : U → U be a linear map. Show that if φ^2 = φ and φ φ∗^ = φ∗^ φ, where φ∗ is the adjoint of φ, then φ is an orthogonal projection. [You may find it useful to prove first that if φ φ∗^ = φ∗^ φ, then φ and φ∗^ have the same kernel.]
(c) Show that given a subspace W of U there exists a unique orthogonal projection τ such that im(τ ) = W. If W 1 and W 2 are two subspaces with corresponding orthogonal projections τ 1 and τ 2 , show that τ 2 ◦ τ 1 = 0 if and only if W 1 is orthogonal to W 2.
(d) Let φ : U → U be a linear map satisfying φ^2 = φ. Prove that one can define a positive definite inner product on U such that φ becomes an orthogonal projection.
Paper 3
