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MATHEMATICAL TRIPOS Part IB
Tuesday 3 June 2003 9 to 12
PAPER 1
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 E be a subset of Rn. Prove that the following conditions on E are equivalent: (i) E is closed and bounded.
(ii) E has the Bolzano–Weierstrass property (i.e., every sequence in E has a subsequence convergent to a point of E).
(iii) Every continuous real-valued function on E is bounded. [The Bolzano–Weierstrass property for bounded closed intervals in R^1 may be assumed.]
2D Methods
Fermat’s principle of optics states that the path of a light ray connecting two points will be such that the travel time t is a minimum. If the speed of light varies continuously in a medium and is a function c(y) of the distance from the boundary y = 0, show that the path of a light ray is given by the solution to
c(y)y′′^ + c′(y)(1 + y′^2 ) = 0 ,
where y′^ = dydx , etc. Show that the path of a light ray in a medium where the speed of light c is a constant is a straight line. Also find the path from (0, 0) to (1, 0) if c(y) = y, and sketch it.
3H Statistics Derive the least squares estimators ˆα and βˆ for the coefficients of the simple linear regression model Yi = α + β(xi − ¯x) + εi, i = 1,... , n,
where x 1 ,... , xn are given constants, ¯x = n−^1
∑n i=1 xi, and^ εi^ are independent with E εi = 0, Var εi = σ^2 , i = 1,... , n.
A manufacturer of optical equipment has the following data on the unit cost (in pounds) of certain custom-made lenses and the number of units made in each order:
No. of units, xi 1 3 5 10 12 Cost per unit, yi 58 55 40 37 22
Assuming that the conditions underlying simple linear regression analysis are met, estimate the regression coefficients and use the estimated regression equation to predict the unit cost in an order for 8 of these lenses.
[Hint: for the data above, Sxy =
∑n i=1(xi^ −^ x¯)yi^ =^ −^257 .4.]
Paper 1
8G Quadratic Mathematics
Let U and V be finite-dimensional vector spaces. Suppose that b and c are bilinear forms on U × V and that b is non-degenerate. Show that there exist linear endomorphisms S of U and T of V such that c(x, y) = b(S(x), y) = b(x, T (y)) for all (x, y) ∈ U × V.
9A Quantum Mechanics
A particle of mass m is confined inside a one-dimensional box of length a. Determine the possible energy eigenvalues.
Paper 1
SECTION II
10F Analysis II
Explain briefly what is meant by a metric space, and by a Cauchy sequence in a metric space.
A function d : X × X → R is called a pseudometric on X if it satisfies all the conditions for a metric except the requirement that d(x, y) = 0 implies x = y. If d is a pseudometric on X, show that the binary relation R on X defined by x R y ⇔ d(x, y) = 0 is an equivalence relation, and that the function d induces a metric on the set X/R of equivalence classes.
Now let (X, d) be a metric space. If (xn) and (yn) are Cauchy sequences in X, show that the sequence whose nth term is d(xn, yn) is a Cauchy sequence of real numbers. Deduce that the function d defined by
d((xn), (yn)) = (^) nlim→∞ d(xn, yn)
is a pseudometric on the set C of all Cauchy sequences in X. Show also that there is an isometric embedding (that is, a distance-preserving mapping) X → C/R, where R is the equivalence relation on C induced by the pseudometric d as in the previous paragraph. Under what conditions on X is X → C/R bijective? Justify your answer.
11D Methods
(a) Determine the Green’s function G(x, ξ) for the operator d
2 dx^2 +^ k
(^2) on [0, π] with
Dirichlet boundary conditions by solving the boundary value problem
d^2 G dx^2
- k^2 G = δ(x − ξ) , G(0) = 0, G(π) = 0
when k is not an integer.
(b) Use the method of Green’s functions to solve the boundary value problem
d^2 y dx^2
- k^2 y = f (x) , y(0) = a, y(π) = b
when k is not an integer.
Paper 1 [TURN OVER
15C Fluid Dynamics
Starting from the Euler equations for incompressible, inviscid flow
ρ
Du Dt
= −∇p, ∇ · u = 0,
derive the vorticity equation governing the evolution of the vorticity ω = ∇ × u.
Consider the flow
u = β(−x, −y, 2 z) + Ω(t)(−y, x, 0),
in Cartesian coordinates (x, y, z), where t is time and β is a constant. Compute the vorticity and show that it evolves in time according to
ω = ω 0 e^2 βtk,
where ω 0 is the initial magnitude of the vorticity and k is a unit vector in the z-direction.
Show that the material curve C(t) that takes the form
x^2 + y^2 = 1 and z = 1
at t = 0 is given later by
x^2 + y^2 = a^2 (t) and z =
a^2 (t)
where the function a(t) is to be determined.
Calculate the circulation of u around C and state how this illustrates Kelvin’s circulation theorem.
16B Complex Methods Sketch the region A which is the intersection of the discs
D 0 = {z ∈ C : |z| < 1 } and D 1 = {z ∈ C : |z − (1 + i)| < 1 }.
Find a conformal mapping that maps A onto the right half-plane H = {z ∈ C : Re z > 0 }. Also find a conformal mapping that maps A onto D 0.
[Hint: You may find it useful to consider maps of the form w(z) = azcz++db .]
Paper 1 [TURN OVER
17G Quadratic Mathematics
(a) Suppose p is an odd prime and a an integer coprime to p. Define the Legendre symbol ( ap ) and state Euler’s criterion.
(b) Compute ( − p^1 ) and prove that ( ab p
a p
b p
whenever a and b are coprime to p.
(c) Let n be any integer such that 1 6 n 6 p − 2. Let m be the unique integer such that 1 6 m 6 p − 2 and mn ≡ 1 (mod p). Prove that
( n(n + 1) p
1 + m p
(d) Find p∑− 2
n=
n(n + 1) p
18A Quantum Mechanics
What is the significance of the expectation value
〈Q〉 =
ψ∗(x) Q ψ(x)dx
of an observable Q in the normalized state ψ(x)? Let Q and P be two observables. By considering the norm of (Q + iλP )ψ for real values of λ, show that
〈Q^2 〉〈P 2 〉 > 14 |〈[Q, P ]〉|^2.
The uncertainty ∆Q of Q in the state ψ(x) is defined as
(∆Q)^2 = 〈(Q − 〈Q〉)^2 〉.
Deduce the generalized uncertainty relation,
∆Q∆P > 12 |〈[Q, P ]〉|.
A particle of mass m moves in one dimension under the influence of the potential 1 2 mω
(^2) x (^2). By considering the commutator [x, p], show that the expectation value of the
Hamiltonian satisfies 〈H〉 > 12 ℏω.
Paper 1
