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NATURAL SCIENCES TRIPOS Part IB
NATURAL SCIENCES TRIPOS Part II (General)
Tuesday 25 May 2004 9.00 to 12.
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, 6A).
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.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
[TURN OVER
1B Let u 1 , u 2 and u 3 be a set of orthogonal, right-handed, curvilinear coordinates, and let e 1 , e 2 and e 3 be the corresponding unit basis vectors such that dr =
i=1 hi^ dui^ ei, where^ r^ is a position vector. Show that
∇φ =
e 1 h 1
∂φ ∂u 1
e 2 h 2
∂φ ∂u 2
e 3 h 3
∂φ ∂u 3
. [4]
Evaluate ∇u 1 and show that
∇ × (φ e 1 ) =
e 2 h 1 h 3
∂u 3
(φh 1 ) −
e 3 h 1 h 2
∂u 2
(φh 1 ). [4]
Evaluate ∇u 2 × ∇u 3 and show that
∇ · (φ e 1 ) =
h 1 h 2 h 3
∂u 1
(φh 2 h 3 ). [4]
Use these expressions to deduce general expressions for ∇ · v and ∇ × v, where v has curvilinear components (v 1 , v 2 , v 3 ). [4]
In cylindrical polar coordinates, (r, θ, z), find ∇ × v for v = (αθrp, r^3 , 0). Find values of p and α for which the vector field v is irrotational, i.e. ∇ × v = 0. [4]
2B Write down the partial differential equation that describes how the temperature, θ(x, t), evolves in time in a one-dimensional bar of length L and thermal diffusivity ν, where x is the spatial coordinate and t is time. [2]
What are the boundary conditions if (a) the ends of the bar are kept at zero temperature or (b) the ends of the bar are insulated? [4]
Assume that the initial temperature at t = 0 is
θ(x, 0) =
x if 0 6 x 6 L/2, L − x if L/ 2 < x 6 L.
Find the solution to the problem, θ(x, t), with boundary conditions (a). Also, find the solution to the problem, θ(x, t), with boundary conditions (b). [12]
Compare the behaviour of θ(x, t) as t → ∞ in (a) and (b). [2]
5A Define what is meant by an orthogonal matrix and a unitary matrix. [4]
Give the properties required to define the scalar product 〈, 〉 in a vector space V over the complex numbers C, and show that the scalar product is antilinear in the first argument, i.e. for vectors u 1 , u 2 and v in V , and complex constants α and β, show that the scalar product obeys
〈αu 1 + βu 2 , v〉 = α∗〈u 1 , v〉 + β∗〈u 2 , v〉. [6]
Find the eigenvalues and eigenvectors of the matrix A, where
A =
1 a 0 a b 0 0 0 c
a, b and c are real positive constants, and a =
b. [10]
6A For the differential equation
y′′^ + p(x)y′^ + q(x)y = 0 ,
state under what conditions a point x = x 0 is
(i) an ordinary point;
(ii) a regular singular point. [4]
If y 1 (x) and y 2 (x) are two solutions of the above equation, define the Wronksian, W (x), of y 1 and y 2. Show that if W 6 = 0 then y 1 and y 2 are linearly independent. [6]
Find power series solutions of the equation
4 xy′′^ + 2(1 − x)y′^ − y = 0
about the point x = 0, giving the indicial equation and suitable recurrence relations for the coefficients. Find the radius of convergence of the solutions. [10]
7B (a) Define a self-adjoint operator for a scalar product of two functions f (x) and g(x) defined by
〈f, g〉 =
∫ (^) b
a
f ∗gw dx,
where w(x) > 0 for a < x < b. [2]
(b) Express the following equation for the function y(x) in Sturm-Liouville form (1 − x^2 )y′′^ − xy′^ + n^2 y = 0.
Find the required boundary conditions for the linear operator acting on y(x) to be self-adjoint over the interval [− 1 , 1]. [7]
(c) Find the eigenvalues, λ, and eigenfunctions, z(x), of the equation
z′′^ + 4z′^ + (4 + λ)z = 0, z(0) = z(1) = 0.
What is the orthogonality relation for these eigenfunctions? [11]
8B Calculate the Green’s function, G(x, ζ), for the following ordinary differential equation x^2 y′′(x) + 2xy′(x) − 2 y(x) = f (x), (∗)
where y(x) is bounded at x = 0 and as x → ∞, and show that it takes the form
G(x, ζ) =
k− xa^ ζb^ if 0 6 x < ζ, k+ ζc^ xd^ if ζ 6 x < ∞,
where a, b, c, d, k− and k+ are constants you need to identify. [7]
Find the solution y(x) of the equation (∗) in terms of G(x, ζ). [2]
Show that if 0 6 f (x) 6 m, then for x > 0
(a) −
m 2
6 y(x) 6 0 ,
and
(b) −
m 3 x
6 y′(x) 6
m 3 x
[11]
[TURN OVER
10C Outline, with justification, why the eigenvalues of the equation,
d dx
p(x)
dy dx
can be found by making
I =
∫ (^) b a (py
′ (^2) + qy (^2) )dx ∫ (^) b a wy
(^2) dx
stationary, where p(x) > 0 and w(x) > 0 for a < x < b and you may assume that y(x) satisfies suitable boundary conditions at x = a and x = b (which should be stated). [9]
A spherically symmetric wavefunction, ψ(r), of the hydrogen atom satisfies for r > 0 the differential equation
r^2 ψ′′^ + 2rψ′^ + 2rψ + 2Er^2 ψ = 0 ,
subject to the boundary conditions ψ(0) = 1 and ψ → 0 as r → ∞. Use the trial function ψ = e−αr^ (α > 0) to estimate the value of the lowest eigenvalue, E. Obtain an alternative estimate using the trial function
ψ =
1 − r/β, 0 < r < β; 0 , r > β.
Which is the better estimate? [11]
[You may quote the results that
∫ (^) ∞
0
sne−sds = n! ,
∫ (^) β
0
s(β − s)^2 ds = 121 β^4 ,
∫ (^) β
0
s^2 (β − s)^2 ds = 301 β^5. ]
[END OF PAPER
