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MATHEMATICAL TRIPOS Part IA
Tuesday 3 June 2003 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. In Section I, you may attempt all four questions. In Section II, at most five answers will be taken into account and no more than three answers on each course will be taken into account.
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 bundles, marked A, B and E according to the code letter affixed to each question. Attach a blue cover sheet to each bundle; write the code letter 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 you have attempted.
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.
SECTION I
1A Algebra and Geometry
The mapping α of R^3 into itself is a reflection in the plane x 2 = x 3. Find the matrix A of α with respect to any basis of your choice, which should be specified.
The mapping β of R^3 into itself is a rotation about the line x 1 = x 2 = x 3 through 2 π/3, followed by a dilatation by a factor of 2. Find the matrix B of β with respect to a choice of basis that should again be specified.
Show explicitly that B^3 = 8A^2
and explain why this must hold, irrespective of your choices of bases.
2B Algebra and Geometry
Show that if a group G contains a normal subgroup of order 3, and a normal subgroup of order 5, then G contains an element of order 15.
Give an example of a group of order 10 with no element of order 10.
3A Vector Calculus
Sketch the curve y^2 = x^2 + 1. By finding a parametric representation, or otherwise, determine the points on the curve where the radius of curvature is least, and compute its value there.
[Hint: you may use the fact that the radius of curvature of a parametrized curve (x(t), y(t)) is ( ˙x^2 + ˙y^2 )^3 /^2 /| x˙¨y − ¨x y˙|.]
4A Vector Calculus
Suppose V is a region in R^3 , bounded by a piecewise smooth closed surface S, and φ(x) is a scalar field satisfying
∇^2 φ = 0 in V, and φ = f (x) on S.
Prove that φ is determined uniquely in V.
How does the situation change if the normal derivative of φ rather than φ itself is specified on S?
Paper 3
6E Algebra and Geometry
Derive an expression for the triple scalar product
e 1 × e 2
· e 3 in terms of the determinant of the matrix E whose rows are given by the components of the three vectors e 1 , e 2 , e 3.
Use the geometrical interpretation of the cross product to show that ea, a = 1, 2 , 3, will be a not necessarily orthogonal basis for R^3 as long as det E 6 = 0.
The rows of another matrix Eˆ are given by the components of three other vectors ˆeb, b = 1, 2 , 3. By considering the matrix E EˆT, where T^ denotes the transpose, show that there is a unique choice of Eˆ such that ˆeb is also a basis and
ea · ˆeb = δab.
Show that the new basis is given by
ˆe 1 =
e 2 × e 3 ( e 1 × e 2
· e 3
etc.
Show that if either ea or ˆeb is an orthonormal basis then E is a rotation matrix.
7B Algebra and Geometry
Let G be the group of M¨obius transformations of C ∪ {∞} and let X = {α, β, γ} be a set of three distinct points in C ∪ {∞}.
(i) Show that there exists a g ∈ G sending α to 0, β to 1, and γ to ∞.
(ii) Hence show that if H = {g ∈ G | gX = X}, then H is isomorphic to S 3 , the symmetric group on 3 letters.
8B Algebra and Geometry
(a) Determine the characteristic polynomial and the eigenvectors of the matrix
Is it diagonalizable?
(b) Show that an n × n matrix A with characteristic polynomial f (t) = (t − μ)n^ is diagonalizable if and only if A = μI.
Paper 3
9A Vector Calculus
Let C be the closed curve that is the boundary of the triangle T with vertices at the points (1, 0 , 0), (0, 1 , 0) and (0, 0 , 1).
Specify a direction along C and consider the integral ∮
C
A · dx ,
where A = (z^2 − y^2 , x^2 − z^2 , y^2 − x^2 ). Explain why the contribution to the integral is the same from each edge of C, and evaluate the integral.
State Stokes’s theorem and use it to evaluate the surface integral ∫
T
(∇ × A) · dS ,
the components of the normal to T being positive.
Show that dS in the above surface integral can be written in the form (1, 1 , 1) dy dz. Use this to verify your result by a direct calculation of the surface integral.
10A Vector Calculus Write down an expression for the Jacobian J of a transformation
(x, y, z) → (u, v, w).
Use it to show that (^) ∫
D
f dx dy dz =
∆
φ |J| du dv dw
where D is mapped one-to-one onto ∆, and
φ(u, v, w) = f
x(u, v, w), y(u, v, w), z(u, v, w)
Find a transformation that maps the ellipsoid D,
x^2 a^2
y^2 b^2
z^2 c^2
onto a sphere. Hence evaluate (^) ∫
D
x^2 dx dy dz.
Paper 3 [TURN OVER
