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EART40005 – MATHS METHODS
Creation date：2024-04-24 16:05:15
Belonging category：数学mathematics
MATHS METHODS

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EART40005 – MATHS METHODS 1
This examination paper comprises 6 questions.
Answer ALL questions.
The total marks for this exam are 50.
This time-limited remote assessment is being run as an open-book examination. We have worked hard to create exams that assesses
synthesis of knowledge rather than factual recall. Thus, access to the internet, notes or other sources of factual information in the
time provided will not be helpful and may well limit your time to successfully synthesise the answers required. Where individual
questions rely more on factual recall and may therefore be less discriminatory in an open book context, we may compare the
performance on these questions to similar style questions in previous years and we may scale or ignore the marks associated with
such questions or parts of the questions. The use of the work of another student, past or present, constitutes plagiarism. Giving
your work to another student to use may also constitute an offence. Collusion is a form of plagiarism and will be treated in a
similar manner. This is an individual assessment and thus should be completed solely by you. The College will investigate all
instances where an examination or assessment offence is reported or suspected, using plagiarism software, vivas and other tools,
and apply appropriate penalties to students. In all examinations we will analyse exam performance against previous performance
and against data from previous years and use an evidence-based approach to maintain a fair and robust examination. As with all
exams, the best strategy is to read the question carefully and answer as fully as possible, taking account of the time and number
of marks available.

– 2 –
Candidates should answer ALL questions.
1. Consider the first-order, ordinary differential equation:
dh
dt
= −h
τ
.
Where τ is a constant.
(a) Using the method of separation of variables, or otherwise, find the function h(t) that
satisfies this equation, given the initial condition that h = −25 when t = 0.
(4 marks)
(b) This differential equation describes, approximately, the rebound of the crust after removal
of an ice sheet; h represents the surface elevation in metres and t represents the time in
years. In this case, describe the meaning of the constant τ .
(2 marks)
2. Consider the matrix equation Ax = b, where A, x and b are given by:
A =
 0 2 41 1 2
2 6 3
 , x =
 xy
z
 , b =
 4022
52
 .
(a) The matrix equation Ax = b represents three linear equations. Write out these equations
explicitly, without using matrix notation. If x, y and z are Cartesian coordinates, what
do these three equations represent?
(2 marks)
(b) Show that the determinant of A, detA = 18. What does this imply about the system of
equations and its geometric interpretation? (3 marks)
(c) Using Cramer’s rule to find at least one variable, solve the system of equations. Show and
explain each step of your calculation.
(5 marks)
3. Consider the infinite power series:
1
1− x = 1 + x+ x
2 + x3 + . . .
(a) By differentiating the power series, or otherwise, find the power series for the function:
1
(1− x)2 .
(2 marks)
(b) Write down an expression for the kth term in the power series you found in Part 3(a).
(2 marks)
(c) For what range of x does the power series you found in Part 3(a) converge? Justify your
answer.
(4 marks)
PAPER CONTINUED ON NEXT PAGE
– 3 –
(0,5)
(3,4)(-4,3)
x
y
(0,0)
A
B
C
Figure 1:
4. Consider the triangle ABC in Figure 1 above.
(a) Convert the Cartesian co-ordinates of A, B and C to Polar co-ordinates. What is signifi-
cant about the position of these points relative to the origin?
(4 marks).
(b) By considering the position vectors of points A, B and C, or otherwise, find the vectors
a =
−→
CA and b =
−−→
CB.
(2 marks).
(c) By calculating a× b or otherwise, calculate the area of triangle ABC.
(2 marks).
5. Evaluate the double integral: ∫ 1
0
∫ 1
0
2xy2exy dx dy.
Clearly show all steps in your solution.
(8 marks)
6. Consider the scalar field φ(x, y, z) = C(x2 + y2 + z2) in a 3D Cartesian coordinate system.
(a) Calculate the gradient vector field ∇φ.
(3 marks)
(b) If C = 23piGρ, where pi, G and ρ are constants, show by finding the divergence of ∇φ, or
otherwise, that φ satisfies the Poisson equation ∇2φ = 4piGρ.
(4 marks)
(c) If φ is the gravitational potential within a uniform spherical body, centred at the origin
(0, 0, 0), and g = −∇φ is the gravitational acceleration vector field, use your answer to
Part 6(a), with constant C = 23piGρ, and your knowledge of spherical coordinates to derive
an equation for how gravitational acceleration varies as a function of radius r inside the
body. Use a sketch to illustrate your answer.
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SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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