Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the Exam of Mathematics which includes Vectors and Matrices, Map, Complex Numbers, Picture, Complex Number, Meant, Dilation, Analysis, Continuous etc. Key important points are: Variational Principles, Analysis, Real Number, Space of Sequences, Norms, Space of Sequences, Lipschitz Equivalent, Considering Sequences, Map, Differentiable
Typology: Exams
1 / 44
This page cannot be seen from the preview
Don't miss anything!
List of Courses
Analysis II
Complex Analysis
Complex Analysis or Complex Methods
Complex Methods
Electromagnetism
Fluid Dynamics
Geometry
Groups Rings and Modules
Linear Algebra
Markov Chains
Methods
Metric and Topological Spaces
Numerical Analysis
Optimization
Quantum Mechanics
Statistics
Variational Principles
Part IB, 2010 List of Questions [TURN OVER
Paper 2, Section I 3G Analysis II Let c > 1 be a real number, and let Fc be the space of sequences a = (a 1 , a 2 ,... ) of real numbers ai with
r=1 c −r (^) |ar| convergent. Show that ‖a‖c = ∑∞ r=1 c −r (^) |ar| defines a norm on Fc.
Let F denote the space of sequences a with |ai| bounded; show that F ⊂ Fc. If c′^ > c , show that the norms on F given by restricting to F the norms ‖. ‖c on Fc and ‖. ‖c′ on Fc′ are not Lipschitz equivalent.
By considering sequences of the form a(n)^ = (a, a^2 ,... , an, 0 , 0 ,... ) in F , for a an appropriate real number, or otherwise, show that F (equipped with the norm ‖. ‖c) is not complete.
Paper 3, Section I 2G Analysis II Consider the map f : R^3 → R^3 given by
f (x, y, z) = (x + y + z, xy + yz + zx, xyz).
Show that f is differentiable everywhere and find its derivative.
Stating carefully any theorem that you quote, show that f is locally invertible near a point (x, y, z) unless (x − y)(y − z)(z − x) = 0.
Paper 4, Section I 3G Analysis II Let S denote the set of continuous real-valued functions on the interval [0, 1]. For f, g ∈ S , set
d 1 (f, g) = sup {|f (x) − g(x)| : x ∈ [0, 1]} and d 2 (f, g) =
0
|f (x) − g(x)| dx.
Show that both d 1 and d 2 define metrics on S. Does the identity map on S define a continuous map of metric spaces (S, d 1 ) → (S, d 2 )? Does the identity map define a continuous map of metric spaces (S, d 2 ) → (S, d 1 )?
Part IB, 2010 List of Questions
Paper 3, Section II 12G Analysis II Let f : U → Rn^ be a map on an open subset U ⊂ Rm. Explain what it means for f to be differentiable on U. If g : V → Rm^ is a differentiable map on an open subset V ⊂ Rp with g(V ) ⊂ U , state and prove the Chain Rule for the derivative of the composite f g.
Suppose now F : Rn^ → R is a differentiable function for which the partial derivatives D 1 F (x) = D 2 F (x) =... = DnF (x) for all x ∈ Rn. By considering the function G : Rn^ → R given by
G(y 1 ,... , yn) = F
y 1 ,... , yn− 1 , yn −
n∑− 1
i=
yi
or otherwise, show that there exists a differentiable function h : R → R with F (x 1 ,... , xn) = h(x 1 + · · · + xn) at all points of Rn.
Paper 4, Section II 12G Analysis II What does it mean to say that a function f on an interval in R is uniformly continuous? Assuming the Bolzano–Weierstrass theorem, show that any continuous function on a finite closed interval is uniformly continuous. Suppose that f is a continuous function on the real line, and that f (x) tends to finite limits as x → ±∞; show that f is uniformly continuous.
If f is a uniformly continuous function on R, show that f (x)/x is bounded as x → ±∞. If g is a continuous function on R for which g(x)/x → 0 as x → ±∞, determine whether g is necessarily uniformly continuous, giving proof or counterexample as appropriate.
Part IB, 2010 List of Questions
Paper 4, Section I 4G Complex Analysis State the principle of the argument for meromorphic functions and show how it follows from the Residue theorem.
Paper 3, Section II 13G Complex Analysis State Morera’s theorem. Suppose fn (n = 1, 2 ,... ) are analytic functions on a domain U ⊂ C and that fn tends locally uniformly to f on U. Show that f is analytic on U. Explain briefly why the derivatives f (^) n′ tend locally uniformly to f ′.
Suppose now that the fn are nowhere vanishing and f is not identically zero. Let a be any point of U ; show that there exists a closed disc ∆ ⊂ U with centre a, on which the convergence of fn and f (^) n′ are both uniform, and where f is nowhere zero on ∆ \ {a}. By considering 1 2 πi
C
f (^) n′(w) fn(w)
dw
(where C denotes the boundary of ∆), or otherwise, deduce that f (a) 6 = 0.
Part IB, 2010 List of Questions [TURN OVER
Paper 2, Section II 13A Complex Analysis or Complex Methods (a) Prove that a complex differentiable map, f (z), is conformal, i.e. preserves angles, provided a certain condition holds on the first complex derivative of f (z). (b) Let D be the region
D := {z ∈ C : |z− 1 | > 1 and |z− 2 | < 2 }.
Draw the region D. It might help to consider the two sets
C(1) := {z ∈ C : |z − 1 | = 1 }, C(2) := {z ∈ C : |z − 2 | = 2 }.
(c) For the transformations below identify the images of D.
Step 1: The first map is f 1 (z) = z − 1 z
Step 2: The second map is the composite f 2 f 1 where f 2 (z) = (z − 12 )i, Step 3: The third map is the composite f 3 f 2 f 1 where f 3 (z) = e^2 πz^.
(d) Write down the inverse map to the composite f 3 f 2 f 1 , explaining any choices of branch. [The composite f 2 f 1 means f 2 (f 1 (z)).]
Part IB, 2010 List of Questions [TURN OVER
Paper 3, Section I 4A Complex Methods (a) Prove that the real and imaginary parts of a complex differentiable function are harmonic. (b) Find the most general harmonic polynomial of the form
u(x, y) = ax^3 + bx^2 y + cxy^2 + dy^3 ,
where a, b, c, d, x and y are real.
(c) Write down a complex analytic function of z = x + iy of which u(x, y) is the real part.
Part IB, 2010 List of Questions
Paper 2, Section I 6C Electromagnetism Write down Maxwell’s equations for electromagnetic fields in a non-polarisable and non-magnetisable medium.
Show that the homogenous equations (those not involving charge or current densit- ies) can be solved in terms of vector and scalar potentials A and φ.
Then re-express the inhomogeneous equations in terms of A, φ and f = ∇·A+c−^2 φ˙. Show that the potentials can be chosen so as to set f = 0 and hence rewrite the inhomogeneous equations as wave equations for the potentials. [You may assume that the inhomogeneous wave equation ∇^2 ψ − c−^2 ψ¨ = σ(x, t) always has a solution ψ(x, t) for any given σ(x, t).]
Paper 4, Section I 7B Electromagnetism Give an expression for the force F on a charge q moving at velocity v in electric and magnetic fields E and B. Consider a stationary electric circuit C, and let S be a stationary surface bounded by C. Derive from Maxwell’s equations the result
dΦ dt
where the electromotive force E =
C q
− (^1) F·dr and the flux Φ = ∫ S B·dS^. Now assume that (∗) also holds for a moving circuit. A circular loop of wire of radius a and total resistance R, whose normal is in the z-direction, moves at constant speed v in the x-direction in the presence of a magnetic field B = (0, 0 , B 0 x/d). Find the current in the wire.
Part IB, 2010 List of Questions
Paper 1, Section II 16C Electromagnetism A capacitor consists of three perfectly conducting coaxial cylinders of radii a, b and c where 0 < a < b < c, and length L where L ≫ c so that end effects may be ignored. The inner and outer cylinders are maintained at zero potential, while the middle cylinder is held at potential V. Assuming its cylindrical symmetry, compute the electrostatic potential within the capacitor, the charge per unit length on the middle cylinder, the capacitance and the electrostatic energy, both per unit length.
Next assume that the radii a and c are fixed, as is the potential V , while the radius b is allowed to vary. Show that the energy achieves a minimum when b is the geometric mean of a and c.
Paper 2, Section II 18C Electromagnetism A steady current I 2 flows around a loop C 2 of a perfectly conducting narrow wire. Assuming that the gauge condition ∇ · A = 0 holds, the vector potential at points away from the loop may be taken to be
A(r) = μ 0 I 2 4 π
C 2
dr 2 |r − r 2 |
First verify that the gauge condition is satisfied here. Then obtain the Biot-Savart formula for the magnetic field B(r) = μ 0 I 2 4 π
C 2
dr 2 × (r − r 2 ) |r − r 2 |^3
Next suppose there is a similar but separate loop C 1 with current I 1. Show that the magnetic force exerted on loop C 1 by loop C 2 is
μ 0 I 1 I 2 4 π
C 1
C 2
dr 1 ×
dr 2 × r 1 − r 2 |r 1 − r 2 |^3
Is this consistent with Newton’s third law? Justify your answer.
Part IB, 2010 List of Questions [TURN OVER
Paper 1, Section I 5B Fluid Dynamics A planar solenoidal velocity field has the velocity potential
φ(x, y, t) = xe−t^ + yet.
Find and sketch (i) the streamlines at t = 0; (ii) the pathline that passes through the origin at t = 0; (iii) the locus at t = 0 of points that pass through the origin at earlier times (streakline).
Paper 2, Section I 7B Fluid Dynamics Write down an expression for the velocity field of a line vortex of strength κ.
Consider N identical line vortices of strength κ arranged at equal intervals round a circle of radius a. Show that the vortices all move around the circle at constant angular velocity (N − 1)κ/(4πa^2 ).
Paper 1, Section II 17B Fluid Dynamics Starting with the Euler equations for an inviscid incompressible fluid, derive Bernoulli’s theorem for unsteady irrotational flow.
Inviscid fluid of density ρ is contained within a U-shaped tube with the arms vertical, of height h and with the same (unit) cross-section. The ends of the tube are closed. In the equilibrium state the pressures in the two arms are p 1 and p 2 and the heights of the fluid columns are ℓ 1 , ℓ 2.
The fluid in arm 1 is displaced upwards by a distance ξ (and in the other arm downward by the same amount). In the subsequent evolution the pressure above each column may be taken as inversely proportional to the length of tube above the fluid surface. Using Bernoulli’s theorem, show that ξ(t) obeys the equation
ρ(ℓ 1 + ℓ 2 )ξ¨ + p 1 ξ h − ℓ 1 − ξ
p 2 ξ h − ℓ 2 + ξ
Now consider the special case ℓ 1 = ℓ 2 = ℓ 0 , p 1 = p 2 = p 0. Construct a first integral of this equation and hence give an expression for the total kinetic energy ρℓ 0 ξ˙^2 of the flow in terms of ξ and the maximum displacement ξmax.
Part IB, 2010 List of Questions [TURN OVER
Paper 3, Section II 18B Fluid Dynamics Write down the exact kinematic and dynamic boundary conditions that apply at the free surface z = η(x, t) of a fluid layer in the presence of gravity in the z-direction. Show how these may be approximated for small disturbances of a hydrostatic state about z = 0. (The flow of the fluid is in the (x, z)-plane and may be taken to be irrotational, and the pressure at the free surface may be assumed to be constant.)
Fluid of density ρ fills the region 0 > z > −h. At z = −h the z-component of the velocity is ǫRe(eiωt^ cos kx), where |ǫ| ≪ 1. Find the resulting disturbance of the free surface, assuming this to be small. Explain physically why your answer has a singularity for a particular value of ω^2.
Paper 4, Section II 18B Fluid Dynamics Write down the velocity potential for a line source flow of strength m located at (r, θ) = (d, 0) in polar coordinates (r, θ) and derive the velocity components ur, uθ.
A two-dimensional flow field consists of such a source in the presence of a circular cylinder of radius a (a < d) centred at the origin. Show that the flow field outside the cylinder is the sum of the original source flow, together with that due to a source of the same strength at (a^2 /d, 0) and another at the origin, of a strength to be determined.
Use Bernoulli’s law to find the pressure distribution on the surface of the cylinder, and show that the total force exerted on it is in the x-direction and of magnitude
m^2 ρ 2 π^2
∫ (^2) π
0
ad^2 sin^2 θ cos θ (a^2 + d^2 − 2 ad cos θ)^2 dθ ,
where ρ is the density of the fluid. Without evaluating the integral, show that it is positive. Comment on the fact that the force on the cylinder is therefore towards the source.
Part IB, 2010 List of Questions
Paper 4, Section II 15F Geometry Suppose that D is the unit disc, with Riemannian metric
ds^2 = dx^2 + dy^2 1 − (x^2 + y^2 )
[Note that this is not a multiple of the Poincar´e metric.] Show that the diameters of D are, with appropriate parametrization, geodesics.
Show that distances between points in D are bounded, but areas of regions in D are unbounded.
Part IB, 2010 List of Questions
Paper 2, Section I 2H Groups Rings and Modules Give the definition of conjugacy classes in a group G. How many conjugacy classes are there in the symmetric group S 4 on four letters? Briefly justify your answer.
Paper 3, Section I 1H Groups Rings and Modules Let A be the ring of integers Z or the polynomial ring C[X]. In each case, give an example of an ideal I of A such that the quotient ring R = A/I has a non-trivial idempotent (an element x ∈ R with x 6 = 0, 1 and x^2 = x) and a non-trivial nilpotent element (an element x ∈ R with x 6 = 0 and xn^ = 0 for some positive integer n). Exhibit these elements and justify your answer.
Paper 4, Section I 2H Groups Rings and Modules Let M be a free Z-module generated by e 1 and e 2. Let a, b be two non-zero integers, and N be the submodule of M generated by ae 1 + be 2. Prove that the quotient module M/N is free if and only if a, b are coprime.
Paper 1, Section II 10H Groups Rings and Modules Prove that the kernel of a group homomorphism f : G → H is a normal subgroup of the group G.
Show that the dihedral group D 8 of order 8 has a non-normal subgroup of order
Part IB, 2010 List of Questions [TURN OVER
Paper 1, Section I 1F Linear Algebra Suppose that V is the complex vector space of polynomials of degree at most n − 1 in the variable z. Find the Jordan normal form for each of the linear transformations
d dz and z d dz
acting on V.
Paper 2, Section I 1F Linear Algebra Suppose that φ is an endomorphism of a finite-dimensional complex vector space.
(i) Show that if λ is an eigenvalue of φ, then λ^2 is an eigenvalue of φ^2.
(ii) Show conversely that if μ is an eigenvalue of φ^2 , then there is an eigenvalue λ of φ with λ^2 = μ.
Paper 4, Section I 1F Linear Algebra Define the notion of an inner product on a finite-dimensional real vector space V , and the notion of a self-adjoint linear map α : V → V. Suppose that V is the space of real polynomials of degree at most n in a variable t. Show that 〈f, g〉 =
− 1
f (t)g(t) dt
is an inner product on V , and that the map α : V → V :
α(f )(t) = (1 − t^2 )f ′′(t) − 2 tf ′(t)
is self-adjoint.
Part IB, 2010 List of Questions [TURN OVER
Paper 1, Section II 9F Linear Algebra Let V denote the vector space of n × n real matrices.
(1) Show that if ψ(A, B) = tr(ABT^ ), then ψ is a positive-definite symmetric bilinear form on V.
(2) Show that if q(A) = tr(A^2 ), then q is a quadratic form on V. Find its rank and signature.
[Hint: Consider symmetric and skew-symmetric matrices.]
Paper 2, Section II 10F Linear Algebra (i) Show that two n × n complex matrices A, B are similar (i.e. there exists invertible P with A = P −^1 BP ) if and only if they represent the same linear map Cn^ → Cn^ with respect to different bases. (ii) Explain the notion of Jordan normal form of a square complex matrix. (iii) Show that any square complex matrix A is similar to its transpose. (iv) If A is invertible, describe the Jordan normal form of A−^1 in terms of that of A. Justify your answers.
Paper 3, Section II 10F Linear Algebra Suppose that V is a finite-dimensional vector space over C, and that α : V → V is a C-linear map such that αn^ = 1 for some n > 1. Show that if V 1 is a subspace of V such that α(V 1 ) ⊂ V 1 , then there is a subspace V 2 of V such that V = V 1 ⊕ V 2 and α(V 2 ) ⊂ V 2.
[Hint: Show, for example by picking bases, that there is a linear map π : V → V 1 with π(x) = x for all x ∈ V 1. Then consider ρ : V → V 1 with ρ(y) = (^1) n
∑n− 1 i=0 α iπα−i(y).]
Part IB, 2010 List of Questions
