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NATURAL SCIENCES TRIPOS Part IB & II (General)
Tuesday 30 May 2006 9 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 is 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.
1B
Let x, y, z be Cartesian co-ordinates, and let Φ be a scalar field.
Define the gradient ∇Φ in Cartesian co-ordinates. Let q 1 , q 2 , q 3 be orthogonal curvilinear co-ordinates. Show that the gradient of Φ in orthogonal curvilinear co-ordinates is
∂q 1
e 1 h 1
∂q 2
e 2 h 2
∂q 3
e 3 h 3
and define the quantities ei and hi which appear in this expression. (^) [8]
Oblate spheroidal co-ordinates (R, θ, φ) are defined by
x = cosh R cos θ cos φ y = cosh R cos θ sin φ z = sinh R sin θ.
Show that the co-ordinate surfaces associated with R, θ, φ intersect at right angles. (^) [8]
Show that for these co-ordinates,
hR = hθ =
sinh^2 R + sin^2 θ, hφ = cosh R cos θ.
[4]
Paper 1
3B
The Fourier transform of a function f is given by
f˜ (k) =
−∞
f (x)e−ikxdx.
Give the expression for the inverse Fourier transform.
Let α > 0 and define for non-negative integer n:
fn(x) =
xne−αx^ x > 0 0 x 6 0
Show that
f˜ 0 (k) = 1 α + ik
and f˜n(k) = n α + ik
f˜n− 1 (k) , n = 1, 2 ,...
Hence compute f˜n(k). (^) [8]
Prove the following identity (Parseval’s theorem) for functions f and g:
∫ (^) ∞
−∞
[f (x)]∗g(x)dx =
2 π
−∞
[ f˜ (k)]∗^ g˜(k)dk ,
where ∗^ denotes complex conjugation. (^) [8]
Hence show that
∫ (^) ∞
−∞
(α^2 + k^2 )n+^
dk =
2 π(2n)! (n!)^2 (2α)^2 n+^
[4]
[In this question all functions may be assumed to be sufficiently integrable so that their Fourier transforms and inverse Fourier transforms are well-defined.]
Paper 1
4B
(i) Let a ∈ Rn^ be a fixed n-component real vector, a 6 = 0. Let A+^ and A−^ be real n × n matrices with components
(A±)ij = δij ± aiaj.
Obtain the eigenvalues of A±^ and describe the corresponding eigenvectors. (^) [4]
Show that A+^ is always invertible and obtain necessary and sufficient conditions for A−^ to be invertible. [4]
(ii) Let b ∈ R^3 be a fixed vector, b 6 = 0. Let B+^ and B−^ be real 3 × 3 matrices with components (B±)ij = δij ± bibj + ijkbk.
By choosing a suitable basis for R^3 , or otherwise, determine the real eigenvalues of B± and the corresponding eigenvectors. (^) [6]
Show that B+^ is invertible and obtain necessary and sufficient conditions for B− to be invertible. [6]
5B
What does it mean for a n × n square matrix to be diagonalizable? (^) [2]
Suppose that A is a n × n square matrix such that Ap^ = 0 for some positive integer p. Show that A has 0 as an eigenvalue. Show also that A cannot be diagonalizable unless A = 0. [6]
Let B and C be the matrices
B =
4 + 2α − 2 − 2 − 4 α 3 α − 3 9 − 6 α 2 + α − 1 − 1 − 2 α
for α ∈ R, and
C =
By considering the characteristic polynomials of B and C, determine whether B and C are diagonalizable. (^) [12]
Paper 1 [TURN OVER
7C Show how to express the eigenvalue equation
d^2 y dx^2
dy dx
(where u(x), v(x), w(x) are real functions and w(x) > 0 for a 6 x 6 b) in Sturm–Liouville form, (^) [6] d dx
[
p(x)
dy dx
]
Suppose that y satisfies the boundary conditions
k 1 y(a) + k 2 y′(a) = 0 , l 1 y(b) + l 2 y′(b) = 0 ,
where k 1 , k 2 , l 1 , l 2 are constants. Show that two eigenfunctions ym, yn, with distinct eigenvalues λm 6 = λn satisfy the orthogonality condition
∫ (^) b
a
r(x)ymyndx =
λm − λn
[
p(b)(y′ n(b)ym(b) − y m′(b)yn(b))
− p(a)(y′ n(a)ym(a) − y′ m(a)yn(a))
]
[4]
Find the eigenfunctions and the values of the eigenvalue λ that satisfy the equation
y′′^ + 2αy′^ + (α^2 + λ)y = 0 ,
where y(0) = y(π) = 0. (^) [6]
Put the equation in Sturm–Liouville form and hence determine the orthogonality condition for the eigenfunctions. (^) [4]
Paper 1 [TURN OVER
8A
By first finding the appropriate Green’s function, solve the differential equation
y′′(x) + k^2 y(x) = f (x) ,
where k is a non-zero real number for the cases:
y(0) = y′(0) = 0 , f (x) = 2k cos kx ,
[10]
and
y(0) = y
( (^) π
6 k
= 0 , f (x) =
k^2 2
[10]
[You may use the identity sin(A ± B) = sin A cos B ± cos A sin B.]
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10C State the Euler equation obtained by minimizing
∫ (^) x 2 x 1 f^ (x, y(x), y
′(x)) dx, with
y(x 1 ) and y(x 2 ) fixed at the boundaries. (^) [3]
If f is not an explicit function of x, show that
y′^
∂f ∂y′^
− f
is constant. [3]
A bead slides down a frictionless wire, starting at rest at x = 0, y = 0 and reaching a point B at x = xB , y = yB after a time t. By considering the bead’s total energy, show that its velocity at any point during its motion is given by v =
2 gy. (^) [4]
Hence, show that the time T [y] taken to reach B depends on the shape of the wire y(x) according to
T [y] =
2 g
∫ (^) xB
0
1 + y′^2 y
dx.
Consider the variational problem that determines the shape of the wire that minimizes the time (the brachistochrone) and show that the quantity y(1 + y′^2 ) is constant for such a shape. Hence, determine the parametric equations of the brachistochrone,
x = c(θ − 12 sin 2θ) , y = c sin^2 θ ,
where c is a constant and θ parameterizes the curve. (^) [5]
Show that if xB = l and yB = 0 the minimum time is equal to
2 πl/g. (^) [5]
END OF PAPER
Paper 1
