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MATHEMATICAL TRIPOS Part IA
Tuesday 5 June 2001 1.30 to 4.
PAPER 3
Before you begin read these instructions carefully.
Each question in Section II carries twice the credit of each question in Section I. You may attempt all four questions in Section I and at most five questions from Section II. In Section II no more than three questions on each course may be attempted.
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, marked A, B, C, D, E, F according to the code letter affixed to each question. Attach a blue cover sheet to each bundle; write the code in the box marked ‘SECTION’ on the cover sheet. Do not tie up questions from Section I and Section II in separate bundles.
You must also complete a green master cover sheet listing all the questions attempted by you.
Every cover sheet must bear your examination number and desk number.
SECTION I
1F Algebra and Geometry
For a 2 × 2 matrix A =
a b c d
, prove that A^2 = 0 if and only if a = −d and
bc = −a^2. Prove that A^3 = 0 if and only if A^2 = 0.
[Hint: it is easy to check that A^2 − (a + d)A + (ad − bc)I = 0.]
2D Algebra and Geometry Show that the set of M¨obius transformations of the extended complex plane C ∪ {∞} form a group. Show further that an arbitrary M¨obius transformation can be expressed as the composition of maps of the form
f (z) = z + a, g(z) = kz and h(z) = 1/z.
3C Vector Calculus For a real function f (x, y) with x = x(t) and y = y(t) state the chain rule for the derivative (^) dtd f (x(t), y(t)).
By changing variables to u and v, where u = α(x)y and v = y/x with a suitable function α(x) to be determined, find the general solution of the equation
x ∂f∂x − 2 y ∂f∂y = 6 f.
4A Vector Calculus
Suppose that
u = y^2 sin(xz) + xy^2 z cos(xz), v = 2xy sin(xz), w = x^2 y^2 cos(xz).
Show that u dx + v dy + w dz is an exact differential.
Show that (^) ∫ (^) (π/ 2 , 1 ,1)
(0, 0 ,0)
u dx + v dy + w dz =
π 2
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9C Vector Calculus
Explain, with justification, how the nature of a critical (stationary) point of a function f (x) can be determined by consideration of the eigenvalues of the Hessian matrix H of f (x) if H is non-singular. What happens if H is singular?
Let f (x, y) = (y − x^2 )(y − 2 x^2 ) + αx^2. Find the critical points of f and determine their nature in the different cases that arise according to the values of the parameter α ∈ R.
10A Vector Calculus
State the rule for changing variables in a double integral.
Let D be the region defined by { 1 /x ≤ y ≤ 4 x when 12 ≤ x ≤ 1, x ≤ y ≤ 4 /x when 1 ≤ x ≤ 2.
Using the transformation u = y/x and v = xy, show that
∫
D
4 xy^3 x^2 + y^2
dxdy =
ln
11B Vector Calculus
State the divergence theorem for a vector field u(r) in a closed region V bounded by a smooth surface S.
Let Ω(r) be a scalar field. By choosing u = c Ω for arbitrary constant vector c, show that (^) ∫
V
∇ Ω dv =
S
Ω dS. (∗)
Let V be the bounded region enclosed by the surface S which consists of the cone (x, y, z) = (r cos θ, r sin θ, r/
- with 0 ≤ r ≤
3 and the plane z = 1, where r, θ, z are cylindrical polar coordinates. Verify that (∗) holds for the scalar field Ω = (a − z) where a is a constant.
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12B Vector Calculus
In R^3 show that, within a closed surface S, there is at most one solution of Poisson’s equation, ∇^2 φ = ρ, satisfying the boundary condition on S
α
∂φ ∂n
where α and γ are functions of position on S, and α is everywhere non-negative.
Show that φ(x, y) = e±lx^ sin ly
are solutions of Laplace’s equation ∇^2 φ = 0 on R^2.
Find a solution φ(x, y) of Laplace’s equation in the region 0 < x < π, 0 < y < π that satisfies the boundary conditions
φ = 0 on 0 < x < π y = 0 φ = 0 on 0 < x < π y = π φ + ∂φ/∂n = 0 on x = 0 0 < y < π φ = sin(ky) on x = π 0 < y < π
where k is a positive integer. Is your solution the only possible solution?
END OF PAPER
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