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NATURAL SCIENCES TRIPOS Part IB & II (General)
Monday 27 May 2002 1.30 to 4.
MATHEMATICS (1)
Before you begin read these instructions carefully:
You may submit answers to no more than six questions. All questions carry the same number of marks.
The approximate number of marks allocated to a part of a question will be indicated in the right hand margin.
Write on one side of the paper only and begin each answer on a separate sheet.
At the end of the examination:
Each question has a number and a letter (for example, 6C).
Answers must be tied up in separate bundles, marked A, B or C according to the letter affixed to each question.
Do not join the bundles together.
For each bundle, a blue cover sheet must be completed and attached to the bundle.
A separate yellow master cover sheet listing all the questions attempted must also be completed.
Every cover sheet must bear your examination number and desk number.
1A In a curvilinear system a point P has coordinates (u, v, w) such that
u^2 =
(s + x), v^2 =
(s − x) and w = z,
where (x, y, z) are the rectangular Cartesian coordinates of P and s =
x^2 + y^2 is its distance from the z-axis. Describe the surfaces u = const, v = const and w = const and sketch the loci of intersections of u^2 = 0, 1, 2 and v^2 = 0, 1, 2 with the x, y plane. (^) [5]
Express x, y and z explicitly in terms of u, v and w in such a way that, when u is defined so that u ≥ 0, y and v have the same sign and the point P is uniquely determined by u, v and w. (^) [4]
Show that u, v and w are orthogonal curvilinear coordinates and find the coeffi- cients hu, hv and hw such that dl, the distance between points (u, v, w) and (u + du, v + dv, w + dw), is given by dl^2 = h^2 udu^2 + h^2 v dv^2 + h^2 wdw^2
in the limit dl → 0. [7]
If φ = φ(u, v) only express ∇^2 φ in terms of derivatives with respect to u and v. (^) [4]
[You may use the following formulae
∇φ =
hu
∂φ ∂u
eu +
hv
∂φ ∂v
ev +
hw
∂φ ∂w
ew
and
∇.A =
huhv hw
∂u
(hv hwAu) +
∂v
(hwhuAv ) +
∂w
(huhv Aw)
.]
Paper 1
4B Suppose A is an n × n matrix:
Prove that An^ = 0 if and only if all eigenvalues of A vanish.
[You may use the fact that a matrix satisfies its own characteristic equation] (^) [5]
By considering the quadratic form Q = xT^ Ax, or otherwise, prove that if the matrix A is symmetric then An^ = 0 implies A = 0. (^) [5]
For the following matrix
T =
1 1 α 1 1 β
find the values of α and β for which all the eigenvalues of T vanish and verify that in that case T^3 = 0. [5]
Consider the transformation x′^ = Tx for three dimensional vectors x and x′. Show that all x′^ are confined to a single plane for any given x.
What is the effect of T^2 and T^3 operating on x? (^) [5]
5B (a) Let A and B be n × n Hermitian matrices (A = A†, B = B†^ ) with distinct eigenvalues. Show that:
(i) H = i (AB − BA) is Hermitian, (^) [4]
(ii) the eigenvectors of A and B are identical if and only if AB = BA, (^) [6]
(iii) the matrix N = A + iB can be diagonalised if and only if NN†^ = N†N. (^) [5]
(b) If C is a unitary matrix and A is Hermitian show that
C−^1 AC
)n has real eigenvalues if n is a positive integer. (^) [5]
[You may quote the properties of the eigenvalue and eigenvectors of Hermitian matrices without proof.]
Paper 1
6C The differential equation
(1 − x^2 )y′′^ − xy′^ + m^2 y = 0 , (∗)
has two linearly independent solutions about the origin of the form y = xσ^
n=0 anx
n
(with a 0 6 = 0).
(a) Is the origin x = 0 an ordinary or singular point of this differential equation? Determine the two appropriate values of σ and find recurrence relations between the an’s for the two cases. [8]
(b) Show that if m is an integer then there always exists a polynomial solution, denoted by Tm(x). How many non-zero terms do these polynomials contain? Find the first four polynomials Tm(x) (i.e. m = 0, 1 , 2 , 3) given the normalization Tm(1) = 1. (^) [7]
(c) Use the ratio test to discuss the convergence of the non-polynomial solutions on the interval − 1 ≤ x ≤ 1. Comment on the relationship of the radius of convergence R to the location of the singular points of the differential equation (∗). (^) [5]
7C (a) Consider the general eigenvalue equation
y′′^ − b(x)y′^ + c(x)y = −λd(x)y ,
subject to homogeneous boundary conditions y(0) = y(1) = 0. Find suitable functions p(x), q(x) and w(x) which enable this equation to be re-expressed in Sturm-Liouville form
− (p(x)y′)′^ + q(x)y = λw(x)y. (^) [4]
(b) Find eigenfunctions and eigenvalues for the equation
y′′^ + λy = 0 , (†)
subject to the boundary conditions y(0) = 0 and y′(π/2) = 0. Determine an appropriate normalization for these eigenfunctions. (^) [5]
Use these to obtain an eigenfunction expansion as a solution of the inhomogeneous equation y′′^ + κy = x ,
subject to the same boundary conditions as in (†) and where κ is a constant (not an eigenvalue). (^) [7]
Hence, by choosing appropriate values for κ and x (or otherwise), show that
π^4 96
[4]
Paper 1 [TURN OVER
9C Consider a set of eigenfunctions yn(x), (with n = 0, 1 , 2.. .) and corresponding eigenvalues λn which satisfy the Sturm-Liouville equation
Ly ≡ − [p(x) y′]′^ + q(x)y = λw(x)y ,
with boundary conditions y(0) = y(1) = 0. Assume that the eigenfunctions are unit
normalized, that is,
0 y
(^2) nwdx = 1.
(a) Given an arbitrary function ˜y satisfying the same boundary conditions, we can define
Fn(x) ≡ Ly˜ − λnw(x)˜y.
Show that for every n we must have
∫ (^1)
0
yn(x)Fn(x)dx = 0. [6]
(b) Consider an approximate trial function ˜y (with ˜y(0) = ˜y(1) = 0) which is close to the lowest eigenfunction y 0 so that we can expand the difference in terms of the remaining eigenfunctions as
y˜ = y 0 +
∑^ ∞
n=
anyn.
where the an’s can be assumed small. Use the Rayleigh-Ritz method with the trial function y˜ to show that the lowest eigenvalue λ 0 can be approximated as
λ˜ 0 = λ^0 +^
n=1 a
2 nλn 1 +
n=1 a^2 n
≈ λ 0 +
∑^ ∞
n=
a^2 n(λn − λ 0 ) ,
where terms of order O(a^4 n) have been neglected. (^) [10]
Discuss the relative error in the approximation ˜y to the lowest eigenfunction y 0 compared with the error in the approximation λ˜ 0 to the lowest eigenvalue λ 0. Is λ˜ 0 lower or higher than λ 0? (^) [4]
Paper 1 [TURN OVER
10C (a) Derive the Euler-Lagrange equation satisfied by the function y(x) which makes the integral
I =
∫ (^) b
a
F [x, y(x), y′(x)]dx
stationary subject to the boundary conditions y(a) = y 1 and y(b) = y 2. (^) [7]
(b) Efficient international airline routes must minimize the distance between two locations on the globe.
Show that the path length between two points A(φ 1 , θ 1 ) and B(φ 2 , θ 2 ) on a unit sphere can be expressed in polar coordinates (φ, θ) as
S =
∫ (^) θ 2
θ 1
1 + sin^2 θ φ′^2 dθ.
where φ′^ = dφdθ. (^) [4]
Hence, use the Euler-Lagrange equation to show that a stationary path satisfies
φ(θ) = ±
∫ (^) θ 2
θ 1
dθ sin θ
k^2 sin^2 θ − 1
where k is a constant. [4]
Integrate this expression to find the extremal solution. Use this to specify the shortest route for a flight setting out from Rome (θ ≈ 45 ◦, φ ≈ 15 ◦) and heading for Singapore (θ ≈ 90 ◦, φ ≈ 105 ◦).
[Hint: In the integration consider the substitution t = cot θ.] (^) [5]
Paper 1
