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MATH50004

BSc, MSci and MSc EXAMINATIONS (MATHEMATICS)

May-June 2021

This paper is also taken for the relevant examination for the

Associateship of the Royal College of Science

Multi-variable Calculus and Differential Equations

SUBMIT YOUR ANSWERS SEPARATE PDFs TO THE RELEVANT DROPBOXES ON

BLACKBOARD (ONE FOR EACH QUESTION) WITH COMPLETED COVERSHEETS WITH

YOUR CID NUMBER, QUESTION NUMBERS ANSWERED AND PAGE NUMBERS PER

QUESTION.

Date: Monday, 10 May 2021

Time: 09:00 to 12:00

Time Allowed: 3 hours

Upload Time Allowed: 45 minutes

This paper has 6 Questions.

Candidates should start their solutions to each question on a new sheet of paper.

Each sheet of paper should have your CID, Question Number and Page Number on the

top.

Only use 1 side of the paper.

Allow margins for marking.

Any required additional material(s) will be provided.

Credit will be given for all questions attempted.

Each question carries equal weight.

1. (a) Consider two vector fields A and B. The vector field B is solenoidal. Use subscript notation

to simplify

(A × ∇) × B − A × curl B.

You may assume the relation εijkεklm = δilδjm − δimδjl. (7 marks)

(b) Determine the constants α, β, γ such that the surfaces

αx2

z − xy2 = β, γxz − y

2 = 0,

intersect orthogonally at the point (x, y, z) = (−2, 2, −1). (7 marks)

(c) Consider the double integral

I =

ZZ

R

(y − x) dx dy

where the finite region R is bounded by the lines

y = x + 1, y = x − 3, y = 2 −

1

3

x, y = 4 −

1

3

x.

Use the substitution

u = y − x, v = y +

1

3

x

to evaluate I. (6 marks)

(Total: 20 marks)

MATH50004 Multivariable Calculus and Differential Equations (2021) Page 2

2. A surface S is described parametrically by

x = h cos θ, y = h sin θ, z = a − h, (0 ≤ h ≤ a, 0 ≤ θ ≤ 2π),

where a is a positive constant.

(a) Express z as a function of x and y. Sketch the surface S. Is this an open or closed surface?

Is it convex? (4 marks)

(b) Show that the unit normal nb to S which has nb · k > 0 can be written in the form

nb =

x i + y j

√

2(x

2 + y

2

)

1/2

+

k

√

2

.

(3 marks)

(c) Find the equation of the tangent plane to S at the location where x = a and y = 0.

(3 marks)

(d) If S is projected onto the x − y plane, what is the shape of the resulting projection?

(2 marks)

(e) Suppose A is the vector field

A = x i .

Calculate

Z

S

A · nb dS

using the projection theorem. (5 marks)

[You may assume that a small areal element in polar coordinates is given by r dr dθ].

(f) Check your answer to (e) using the divergence theorem. (3 marks)

(Total: 20 marks)

MATH50004 Multivariable Calculus and Differential Equations (2021) Page 3

3. (a) The coordinates (u1, u2, u3) are defined in terms of Cartesian coordinates (x, y, z) by

x = u1u2 cos u3, y = u1u2 sin u3, z =

1

2

(u

2

1 − u

2

2

), (u1 ≥ 0, u2 ≥ 0, 0 ≤ u3 ≤ 2π).

(i) By calculating an appropriate Jacobian, find the function F(u1, u2) such that an element

of the surface u3 = constant can be expressed as

dS = F(u1, u2) du1du2.

(3 marks)

(ii) Show that this system is orthogonal, determine the lengthscales h1, h2, h3 and verify that

F(u1, u2) = h1h2.

(6 marks)

(b) A curve y = y(x) joins the points (−a, 0),(a, 0) in the x − y plane where a > 0.

(i) What properties of the curve do the integrals

I =

Z a

−a

y

1 +

dy

dx

!2

1/2

dx, J =

Z a

−a

1 +

dy

dx

!2

1/2

dx

represent? What does the ratio I/J represent physically? (3 marks)

(ii) Using the Euler-Lagrange equation show that the appropriate form of y(x) which renders

I stationary subject to the constraint J = 2 is

y = C

cosh

x

C

− cosh

a

C

,

where

C sinh

a

C

= 1.

(6 marks)

(iii) Deduce that solutions can only exist if a < a0 where a0 is a value to be identified.

(2 marks)

(Total: 20 marks)

MATH50004 Multivariable Calculus and Differential Equations (2021) Page 4

4. (a) Consider an initial value problem

x˙ = f(t, x), x(t0) = x0 ,

where f : R × R

d → R

d

is continuous and (t0, x0) ∈ R × R

d

is fixed. Let J be an interval

containing t0 in its interior, and consider the Picard iterates {λn : J → R

d}n∈N0

corresponding

to this initial value problem.

(i) Show that λ˙

n(t0) = f(t0, λn(t0)) for any n ∈ N. (3 marks)

(ii) What is the maximal (i.e. largest) interval J on which the functions λn : J → R

d

can

be defined? Justify your answer. (3 marks)

(iii) Compute λ0, λ1 and λ2 for the one-dimensional initial value problem x˙ = x

2 with

x(1) = 1. (6 marks)

(b) Consider an autonomous differential equation

x˙ = f(x),

where f : R

d → R

d

is locally Lipschitz continuous.

(i) Does this differential equation have unique local solutions for every initial condition of

the form x(0) = x0, where x0 ∈ R

d

? Justify your answer. (2 marks)

(ii) Prove that for all y0 ∈ R

d

, there exist T > 0 and x0 ∈ R

d

such that there exists a

solution λ : I → R

d

to this differential equation with λ(0) = x0 and λ(T) = y0.

(6 marks)

(Total: 20 marks)

MATH50004 Multivariable Calculus and Differential Equations (2021) Page 5

5. Consider the nonlinear differential equation

x˙

y˙

=

2 −2

1 0

| {z }

=:A

x

y

+

−x

(x − y)

2 + y

2

−y

(x − y)

2 + y

2

,

whose right hand side is written as the sum of the linear part with coefficient matrix A and a

nonlinearity.

(i) Show that (x

∗

, y∗

) := (0, 0) is the only equilibrium. (3 marks)

(ii) Calculate the real Jordan normal form of the coefficient matrix A using an invertible

transformation matrix T ∈ R

2×2

. (4 marks)

(iii) Explain why the equilibrium (x

∗

, y∗

) = (0, 0) is repulsive. (2 marks)

(iv) Show that the set MR := {(x, y) ∈ R

2

: (x − y)

2 + y

2 ≤ R} is positively invariant for some

R > 0. (6 marks)

Hint. Consider the orbital derivative of an appropriate scalar-valued function and note that it

is helpful to preserve/create terms of the form ((x − y)

2 + y

2

) in your calculations.

(v) Prove that there exists a periodic orbit. (5 marks)

(Total: 20 marks)

MATH50004 Multivariable Calculus and Differential Equations (2021) Page 6

6. (a) Decide for each of the following four statements whether it is true or false. All statements

involve omega limit sets ω(x) or alpha limit sets α(x) of a differential equation

x˙ = f(x),

where we require that f : R

d → R

d

is locally Lipschitz continuous. Justify your answer

by either providing an example (which can also be a picture with short explanation) or an

explanation why such an example does not exist.

(i) There exist f : R

d → R

d

and x ∈ R

d

such that ω(x) is a singleton. (3 marks)

(ii) There exist f : R

d → R

d

, x ∈ R

d

and y ∈ ω(x) such that ω(x) is nonempty and

compact and ω(x) ∩ ω(y) = ∅. (3 marks)

(iii) There exist f : R

d → R

d

and x ∈ R

d

such that ω(x) = α(x). (3 marks)

(iv) There exist f : R

d → R

d

and x ∈ R

d

such that ω(x) = α(x) and x /∈ ω(x). (3 marks)

(b) Consider an autonomous differential equation

x˙ = f(x),

where f : R

d → R

d

is locally Lipschitz continuous. The flow of this differential equation is

denoted by ϕ, and let x ∈ R

d

such there exists a K > 0 with kϕ(t, x)k ≤ K for all t ≥ 0.

(i) Show that ω(x) is nonempty. (2 marks)

(ii) Show that for all ε > 0, there exists a T > 0 such that

ϕ(t, x) ∈ Bε(ω(x)) for all t ≥ T ,

where Bε(ω(x)) := {y ∈ R

d

: ky − zk < ε for some z ∈ ω(x)}. (6 marks)

(Total: 20 marks)

MATH50004 Multivariable Calculus and Differential Equations (2021) Page 7