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MATHEMATICAL TRIPOS Part IA
Tuesday 1st June 2004 1.30 to 4.
PAPER 3
Before you begin read these instructions carefully.
Each question in Section II carries twice the number of marks 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.
Additional credit will be awarded for substantially complete answers.
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 C and D according to the code letter affixed to each question. Include in the same bundle questions from Sections I and II with the same code letter.
Attach a gold cover sheet to each bundle; write the code letter in the box marked ‘EXAMINER LETTER’ on the cover sheet.
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
1D Algebra and Geometry
State Lagrange’s Theorem. Show that there are precisely two non-isomorphic groups of order 10. [You may assume that a group whose elements are all of order 1 or 2 has order 2k.]
2D Algebra and Geometry
Define the M¨obius group, and describe how it acts on C ∪ {∞}.
Show that the subgroup of the M¨obius group consisting of transformations which fix 0 and ∞ is isomorphic to C∗^ = C \ { 0 }.
Now show that the subgroup of the M¨obius group consisting of transformations which fix 0 and 1 is also isomorphic to C∗.
3C Vector Calculus
If F and G are differentiable vector fields, show that
(i) ∇ × (F × G) = F(∇ · G) − G(∇ · F) + (G · ∇)F − (F · ∇)G , (ii) ∇(F · G) = (F · ∇)G + (G · ∇)F + F × (∇ × G) + G × (∇ × F).
4C Vector Calculus Define the curvature, κ, of a curve in R^3.
The curve C is parametrised by
x(t) =
et^ cos t,
et^ sin t,
et
for − ∞ < t < ∞.
Obtain a parametrisation of the curve in terms of its arc length, s, measured from the origin. Hence obtain its curvature, κ(s), as a function of s.
Paper 3
8D Algebra and Geometry
Compute the characteristic polynomial of
A =
0 4 − s 2 s − 2 0 − 2 s + 2 4 s − 1
Find the eigenvalues and eigenvectors of A for all values of s. For which values of s is A diagonalisable?
9C Vector Calculus
For a function f : R^2 → R state if the following implications are true or false. (No justification is required.)
(i) f is differentiable ⇒ f is continuous.
(ii)
∂f ∂x
and
∂f ∂y
exist ⇒ f is continuous.
(iii) directional derivatives
∂f ∂n
exist for all unit vectors n ∈ R^2 ⇒ f is differentiable.
(iv) f is differentiable ⇒
∂f ∂x
and
∂f ∂y
are continuous.
(v) all second order partial derivatives of f exist ⇒
∂^2 f ∂x ∂y
∂^2 f ∂y ∂x
Now let f : R^2 → R be defined by
f (x, y) =
xy(x^2 − y^2 ) (x^2 + y^2 )
if (x, y) 6 = (0, 0) , 0 if (x, y) = (0, 0).
Show that f is continuous at (0, 0) and find the partial derivatives
∂f ∂x
(0, y) and ∂f ∂y
(x, 0). Then show that f is differentiable at (0, 0) and find its derivative. Investigate
whether the second order partial derivatives
∂^2 f ∂x ∂y
(0, 0) and
∂^2 f ∂y ∂x
(0, 0) are the same.
Are the second order partial derivatives of f at (0, 0) continuous? Justify your answer.
Paper 3
10C Vector Calculus
Explain what is meant by an exact differential. The three-dimensional vector field F is defined by
F =
exz^3 + 3x^2 (ey^ − ez^ ), ey^ (x^3 − z^3 ), 3 z^2 (ex^ − ey^ ) − ez^ x^3
Find the most general function that has F · dx as its differential.
Hence show that the line integral ∫ (^) P 2
P 1
F · dx
along any path in R^3 between points P 1 = (0, a, 0) and P 2 = (b, b, b) vanishes for any values of a and b.
The two-dimensional vector field G is defined at all points in R^2 except (0, 0) by
G =
−y x^2 + y^2
x x^2 + y^2
(G is not defined at (0, 0).) Show that
∮
C
G · dx = 2π
for any closed curve C in R^2 that goes around (0, 0) anticlockwise precisely once without passing through (0, 0).
11C Vector Calculus
Let S 1 be the 3-dimensional sphere of radius 1 centred at (0, 0 , 0), S 2 be the sphere of radius 12 centred at ( 12 , 0 , 0) and S 3 be the sphere of radius 14 centred at ( − 41 , 0 , 0). The eccentrically shaped planet Zog is composed of rock of uniform density ρ occupying the region within S 1 and outside S 2 and S 3. The regions inside S 2 and S 3 are empty. Give an expression for Zog’s gravitational potential at a general coordinate x that is outside S 1. Is there a point in the interior of S 3 where a test particle would remain stably at rest? Justify your answer.
Paper 3 [TURN OVER
