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NATURAL SCIENCES TRIPOS Part IB
NATURAL SCIENCES TRIPOS Part II (General)
Friday 28 May 2004 9.00 to 12.
MATHEMATICS (2)
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.
You may not start to read the questions
printed on the subsequent pages until
instructed to do so by the Invigilator.
[TURN OVER
1C In plane polar coordinates, (r, θ), Poisson’s equation for a potential ϕ(r, θ) generated by a source ρ(r, θ) is
r
∂r
r
∂ϕ ∂r
r^2
∂^2 ϕ ∂θ^2
= ρ.
(a) Using separation of variables derive the general solution of Poisson’s equation that is single-valued and finite in the domain 0 6 r 6 a if
ρ(r, θ) =
∑^ ∞
n=
αnrn^ cos θ ,
where the αn are known constants. [11]
Find the particular solution if ϕ = sin 2θ on r = a for the special case when α 1 = 1 and αn = 0 if n 6 = 1. [4]
(b) The velocity potential ϕ(r, θ) for inviscid flow in two dimensions satisfies Laplace’s equation (i.e. Poisson’s equation with ρ = 0). Assuming that ϕ is single-valued, solve for ϕ(r, θ) in r > a subject to the boundary conditions
∂ϕ ∂r
= 0 on r = a ,
and ∂ϕ ∂r
→ U cos θ as r → ∞ ,
where U is a constant. [5]
3B (a) By writing z = x + iy, derive the Cauchy-Riemann equations for the real and imaginary parts of an analytic function f (z) = u(x, y) + iv(x, y) in Cartesian coordinates (x, y), where the functions u and v are real. [4]
Show that the function g(z) = z Re(z) is differentiable only at z = 0, and find g′(0). [4]
(b) By writing z = reiθ, or otherwise, derive the Cauchy-Riemann equations for the real and imaginary parts of an analytic function f (z) = U (r, θ) + iV (r, θ) in polar coordinates (r, θ), where the functions U and V are real. [5]
(c) Calculate the value of the integral
I =
γ
log z dz ,
where γ is a positively oriented closed contour, in the cases when
(i) γ is a unit circle, the branch cut is taken along the positive real axis, and log z is real on the ‘upper’ side of the branch cut (i.e. log z is real when θ = 0+); (ii) γ is a circle of radius R, the branch cut is taken along the positive imaginary axis, and log z is real on the positive real axis. [7]
[You may quote the result that ∫ θ eiθ^ dθ = (1 − iθ) eiθ^. ]
4B State the residue theorem for the integral of a function of a complex variable around a closed contour. [2]
The function f (z) has the form
f (z) =
φ(z) ψ(z)
in a neighbourhood of a simple pole z = a, where φ(z) and ψ(z) are analytic at z = a, φ(a) 6 = 0, ψ(a) = 0 and ψ′(a) 6 = 0. Show that the residue of f (z) at z = a is given by φ(a)/ψ′(a). Use this result to evaluate the residue of cotan z at z = kπ, where k is an integer. [6]
Evaluate, using the residue theorem, the integral
I =
0
log x x^2 + a^2
dx, a > 0.
What is the value of this integral at a = 1? [12]
[Hint: You may find it useful to consider a large semicircular contour.]
5C Define the convolution of the functions f (t) and g(t) on the assumption that the functions vanish for t < 0. Derive an expression for the Laplace transform of the convolution in terms of the Laplace transforms of f (t) and g(t). [5]
For t > 0 the function y(t) satisfies the equation
d^2 y dt^2
Using Laplace transforms and the convolution theorem show that the general solution to this equation is
y(t) = y(0) cos t +
dy dt
(0) sin t +
∫ (^) t
0
f (τ ) sin(t − τ )dτ ,
where Jordan’s lemma may be used without proof as long as you demonstrate that the conditions of the lemma hold. Confirm that this solution satisfies both the equation, and the initial conditions at t = 0. [13]
Evaluate the solution in the special case when f (t) = δ(t − T ), where δ(t) is the delta function and T > 0. [2]
6B Define an isotropic tensor. Write down the general forms of all non-zero isotropic tensors of ranks 1, 2, and 3. [3]
A vector field ui has the following components in a particular system of Cartesian coordinates xi:
u 1 = x 1 x^22 , u 2 = x 2 x^23 , u 3 = x 3 x^21.
Express the tensor ∂ui/∂xk as a linear combination of ǫijkωj (where ωj is a vector to be determined) and a symmetric tensor eik. [9]
Find the directions of the principal axes of eik at the point x 1 = 2, x 2 = 3 and x 3 = 0, and determine the corresponding principal values. [8]
[Note: suffix notation and the summation convention are assumed in this question.]
[TURN OVER
10A Explain what is meant by a representation D of a group G, and define a faithful representation. [3]
Give a faithful 2-dimensional representation of the group C 4 = { 1 , i, − 1 , −i}, and give a geometrical interpretation. [4]
If D is an n-dimensional representation of a group G, and S is an invertible n × n matrix, show that the map D˜ defined by D˜(g) = SD(g)S−^1 is also a representation, where g ∈ G. [4]
Representations D and D˜ above are equivalent. Define the characters of a representation, and show that
(i) characters of equivalent representations are the same, and
(ii) elements of the same conjugacy class have the same character. [9]
[END OF PAPER
