Fields Forms and Flows 34 Exam 2020 PDF

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UNIVERSITY OF BRISTOL School of Mathematics Fields, Forms and Flows 34 MATH M0033 (Paper code MATH–M0033J)

January 2020 2 hours 30 minutes

This paper contains FOUR questions. All answers will be used for assessment. Calculators are not permitted in this examination.

On this examination, the marking scheme is indicative and is intended only as a guide to the relative weighting of the questions.

Unless stated otherwise, all functions, vector fields, differential forms and maps are taken to be smooth. LX = iX d + diX (F∗ X) (F (x)) = F ′ (x) · X(x) ∂c =

k X X

(−1)j+αc(j,α)

j=1 α=0,1

iX (α ∧ β) = (iX α) ∧ β + (−1)k α ∧ (iX β), where α is a k-form.

Do not turn over until instructed. Page 1 of 5

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1. (a) Let X(x, y) = (2y, 2x) and Y(x, y) = (x, x + y) be vector fields on R2 . (i) (3 marks) Find the flow Φt of X. (ii) (3 marks) Verify that the flow of Y is given by Ψs (x, y) = (x exp(s), xs exp(s) + y exp(s)) .

(1)

(b) Let F : R2 → R2 be the diffeomorphism that is defined by the flow in eq. (1), F (x, y ) = Ψs (x, y ), where s is a parameter. (i) (5 marks) Find F −1 (u, v) and compute (F∗ X)(u, v) for the vector field X(x, y) = (2y, 2x). (ii) (5 marks) Find the set of initial conditions (u0 , v0 ) for which the solutions (u(t), v (t)) of the differential equations, that depend on the parameter s, du = −2su + 2v , dt

dv = 2(1 − s2 )u + 2sv , dt

go to (0, 0) as t → ∞. (c) (3 marks) Let V and W be two vector fields on Rn and let Γt be the flow of W. You are given that   ∂  Γt∗ V = [V, W] . ∂t t=0 Show that

∂ Γt∗ V = [Γt∗ V, W] . ∂t

(d) (6 marks) Let G be a diffeomorphism on Rn and define vector fields A and B by Ai (x) =

∂Gi −1 (G (x)) , ∂x1

Bi (x) =

∂Gi −1 (G (x)) . ∂x2

Show that [A, B] = 0.

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2. (a) (3 marks) Let σ be the permutation given by   1 2 3 4 , 2 4 1 3 i.e. σ(1) = 2, σ(2) = 4, etc. Write σ as a product of transpositions and determine whether it is even or odd. (b) Consider the system of first-order partial differential equations given by ∂u 8xy 2 =− + 6y, ∂x u ∂u 8x2 y =− + 6x, ∂y u with initial data u(x0 , y0 ) = u0 . (i) (4 marks) By appealing to the Frobenius theorem, or otherwise, show that the system has a unique solution u(x, y) in some neighbourhood of (x0 , y0 ) if u0 6= 0. (ii) (11 marks) Find the solution which satisfies u(0, 0) = 1. Your answer may be given in the form of a quadratic equation: u2 + f (x, y)u + g(x, y) = 0 where f and g are functions that you are to determine. (Hint: first-order homogeneous differential equations of the form dz/dt = f (z/t) can be transformed into a separable differential equation by making the substitution v = z/t.) (c) (7 marks) Let u(x, y) and v(x, y) be the real and imaginary parts of an analytic function h(x + iy), so that u and v satisfy the Cauchy-Riemann equations ∂v ∂u = , ∂x ∂y

∂u ∂v =− . ∂x ∂y

(2)

Consider the system of first-order partial differential equations for u(x, y) given by ∂u = r(u, v ), ∂x ∂u = −s(u, v), ∂y

(3)

where r(u, v) and s(u, v) are smooth, with initial data u(x0 , y0 ) = u0 , v(x0 , y0 ) = v0 . Show that u(x, y) and v(x, y) satisfying (2) and (3) exist in a neighbourhood of (x0 , y0 ) if and only if ∂r ∂s ∂r ∂s = , =− . ∂u ∂v ∂v ∂u

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ˆ t is a one-parameter 3. (a) You are given the following statement of the Poincar´e Lemma: If Φ n ˆ family of diffeomorphisms on U ⊂ R and Xt is the time-dependent vector field defined by ˆ t, ˆt ◦ Φ ˆt = ∂ Φ X ∂t and if β is a closed k-form on U such that ˆ ∗1 β = β, Φ

ˆǫ∗β = 0, lim Φ

ǫ→0

then β = dα, where α=

Z

1

0

ˆ ∗(i ˆ β) dt. Φ t Xt

(4)

(i) (5 marks) Let β be the 2-form on R3 − {0} given by β=z

x dy ∧ dz + y dz ∧ dx + z dx ∧ dy (x2 + y 2 + z 2 )2

(5)

Show that dβ = 0. (ii) (3 marks) Let U = {(x, y, z)|z 2 6= 0}, and let Φˆt : U → U be given by ˆ t (x, y, z) = (tx, ty, z) . Φ ˆ ∗1 β and show that Compute Φ ˆΦ1∗β = β,

ˆ ǫ∗β = 0, lim Φ

ǫ→0

(iii) (9 marks) Given the preceding results, find a 1-form α on U such that β = dα. (b) (8 marks) Let ω be a 1-form on Rn . Show that dω(X, Y) = LX (ω(Y)) − LY (ω (X)) − ω ([X, Y]) for any smooth vector fields X, Y. You may use without proof that L[X,Y] = LX LY − LY LX . (Hint: It may help to assume ω to be of a particular form, but you should justify any such assumption.)

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4. (a) Let f be the zero-form on R3 and α the one-form on R3 given by f = xy, α = xy dx + xz dz, and let X be the vector field on R3 given by X = (0, −z, x). (i) (3 marks) Compute df ∧ α, combining terms where possible. (ii) (3 marks) Compute LX α, combining terms where possible. (b) Let c: I 2 → R3 be the singular 2-cube given by   πt πs πt πt πs sin , sin sin , cos . c(s, t) = cos 2 2 2 2 2 Let (x, y, z) denote Cartesian coordinates on R3 , and let ω be the one-form on R3 given by ω = x dy. i. (3 marks) Compute c∗ ω . ii. (3 marks) Show that c∗ dω is given by c∗ dω = −

iii. (2 marks) Compute

R

c

π2 sin(πt) ds ∧ dt. 8

dω.

iv. (4 marks) Without using Stokes’ theorem, compute

R

∂c

ω.

(c) (7 marks) Let f (x, y, z ) = x2 + y 2 + z 2 . Suppose that β is a two-form on R3 such that β ∧ df = 0 . Show that c∗ β = 0 . where c is the singular 2-cube from part (b).

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