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NATURAL SCIENCES TRIPOS Part IB & II (General)
Monday 26 May 2003 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, 6B).
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.
1B Elliptic cylindrical coordinates (u, v, z) are defined in terms of the Cartesian coordinates (x, y, z) by x = a cosh u cos v,
y = a sinh u sin v,
z = z,
where u ≥ 0, 0 ≤ v < 2 π, −∞ < z < ∞. Sketch the surfaces of constant u and the surfaces of constant v. Show that the coordinate surfaces intersect at right angles and thus these coordinates are orthogonal coordinates. (^) [8]
Find the metric coefficients (hu, hv , hz ) defined by
|dr|^2 = dx^2 + dy^2 + dz^2 = h^2 udu^2 + h^2 v dv^2 + h^2 z dz^2 ,
and show that hu = hv. (^) [4]
Obtain the divergence, curl and Laplace’s equation in the coordinates (u, v, z). (^) [8]
2B A bar of length L made of material with diffusivity ν, is initially at a temperature of 0◦C. One end of the bar (x = 0) is held at 0◦C and the other is supplied with heat at a constant rate per unit area H, so that
k
∂u ∂x
(L, t) = H ,
where u(x, t) is the temperature distribution and k is the thermal conductivity.
(i) State the equation with the appropriate boundary and initial conditions for the temperature distribution within the bar after a time t. (^) [3]
(ii) Transform the problem into the equivalent homogeneous one by writing
u(x, t) = v(x, t) + w(x) ,
where w(x) satisfies the equation for the temperature distribution and its boundary conditions (but not the initial condition). (^) [4]
(iii) Solve the resulting problem for v(x, t) using the method of separation of variables. [13]
Paper 1
5B Prove that the eigenvalues of an Hermitian matrix are real. Prove also that the eigenvectors corresponding to distinct eigenvalues of an Hermitian matrix are orthogonal. [7]
Prove that the eigenvalues of a unitary matrix U have unit modulus. (^) [3]
Suppose that a unitary matrix U is written as A + iB, where A and B are symmetric real matrices with non-degenerate eigenvalues.
(a) Show that A and B commute and A^2 + B^2 = 1. (^) [4]
(b) Show that the eigenvectors of A are also eigenvectors of B, and give the eigenvalues of A and B in terms of the eigenvalues of U. (^) [6]
6B Define the terms ordinary point and regular singular point for a second order linear differential equation, and explain briefly the reason for distinguishing between them. (^) [4]
Find power series solutions about z = 0 of
y′′^ − 2 zy′^ + λy = 0. (^) [8]
For what values of λ does the equation possess a polynomial solution? Find the solutions corresponding to two eigenvalues λ of your choice. (^) [8]
Paper 1
7A Consider the eigenvalue problem
y′′^ +
x
y′^ + λy = 0 ,
where the function y(x) satisfies the boundary conditions that y(0) is finite and y(π) = 0. By expressing the equation in self-adjoint form, or otherwise, show that eigenfunctions u(x) and v(x) corresponding to distinct eigenvalues satisfy the orthogonality relation
∫ (^) π
0
x^2 u(x)v(x) dx = 0. [5]
By using the change of dependent variable z = xy, or otherwise, find the set of eigenfunctions and corresponding eigenvalues. (^) [6]
Show that the solution y(x) satisfying the equation
y′′^ +
x
y′^ − βy = 1 , where β > 0 ,
and the boundary conditions that y(0) is finite and y(π) = 0, has the form
y(x) = 2
∑^ ∞
n=
(−1)n n(n^2 + β)
sin nx x
[9]
8A Find the values of p for which y = xp^ is a solution of
( d^2 dx^2
x
d dx
x^2
y = 0. [2]
The function y(x) is defined on the range x ≥ 0 and satisfies the differential equation (^) ( d^2 dx^2
x
d dx
x^2
y = e−x^ , (∗)
and the boundary conditions y(0) = 0, and y(x) → 0 as x → ∞. Find the Green’s function for this problem. (^) [7]
Use it to obtain the solution to equation (∗). (^) [11]
Paper 1 [TURN OVER
10C For functions y(x), p(x), q(x) and w(x) defined on a ≤ x ≤ b, let
F [y] =
∫ (^) b
a
py′^2 + qy^2
dx , G[y] =
∫ (^) b
a
wy^2 dx ,
and
Λ[y] =
F [y] G[y]
On the assumption that p > 0 and w > 0 for a < x < b, and subject to suitable boundary conditions which should be stated, show that the stationary values of Λ[y] are the eigenvalues, λ, of the Sturm-Liouville differential equation
d dx
p(x)
dy dx
Show also that the functions for which Λ[y] is stationary are the corresponding eigenfunc- tions. [7]
Consider now the choice p = 1, q = 0 and w = x (for 0 ≤ x ≤ π), so that y(x) satisfies the differential equation
d^2 y dx^2
By use of a suitable polynomial trial function find an estimate of the lowest eigenvalue, λ 0 , for the choice of boundary conditions y(0) = y(π) = 0. Also find an estimate of the lowest eigenvalue using a suitable trigometric trial function. (^) [10]
Indicate, with reasoning, whether these estimates are greater or less than λ 0 and why you might expect these estimates to be close to each other. (^) [3]
Paper 1
