Student
number
Semester 2 Assessment, 2024
School of Mathematics and Statistics
MAST30031 Methods of Mathematical Physics
Submission deadline: 23:59 on Monday 7 October 2024
This assignment consists of 4 pages (including this page) with 5 questions and 60 total marks
Instructions to Students
Writing
Expectations on the presentation
–Begin the answer to a new question on a new page.
–Marks may be deducted in every question for incomplete working, incorrect use of
mathematical notation and insufficient justification of steps.
–It is usually not sufficient to only write mathematical formulas or symbols. Your
answer should contain text that clearly explains what is going on. You are allowed
(and encouraged) to use results from lecture slides, exercises or earlier assignments
but these need to be named or referenced appropriately. Lack of logical coherence will
lead to the deduction of marks even if the computation or idea is generally correct.
Late or missed assignments
–Students unable to submit assignments due to illness or other extenuating circum-
stances may receive special consideration, provided that the circumstances are signif-
icant and supported by appropriate documentation. Please see the LMS for details.
–Assignments submitted after the due date will not be marked except where an ex-
tension has been granted.
Concerning plagiarism
–Please familiarise yourself with the University’s expectations on Academic Integrity.
Note, in particular: While group discussions about general aspects of the lecture and
exercise problems are generally encouraged, you are expected to submit work that is
original and solely your work.
–Detected plagiarism or other forms of academic misconduct may lead to serious
disciplinary consequences for students involved.
Write your answers on A4 paper. Each question should be on a new page. The question
number must be written at the top of each page.
Scanning and Submitting
Put the pages in question order and all the same way up. Use a scanning app to scan all
pages to PDF. Scan directly from above. Crop pages to A4.
Submit your scanned assignment as a single PDF file and carefully review the submission
in Gradescope. Scan again and resubmit if necessary.
Make sure to assign pages to questions appropriately within Gradescope.
©University of Melbourne 2024Page 1 of 4 pagesDo not place in Baillieu Library
Question 1 (10 marks)
Write a summary of the first part of the lectures called “Differential Forms” which is covered
in Weeks 7-9.The summary should be between two and three A4 pages!To make it
simple, pickten out of the fourteen topics belowand briefly describe those in a concise
way.
•differential 0-forms and 1-forms as vector spaces
•exact and closed 1-forms
•wedge product as multilinear maps
•exterior derivative
•closed and exact p-forms
•wedge product and grad, div, curl,
•integrating p-forms (higher dimensional contour integrals)
•orientations and differential p-forms
•boundaries and their orientations
•generalised Stoke’s theorem
•Hodge star operator
•the Laplacian for differential forms
•de Rahm cohomologies
•Maxwell’s equations
Question 2 (15 marks)[This question is from a previous exam]
LetL={(x,y,z)∈R
3
: (x−z)
2
+ (y−z)
2
= 0}. Consider the two differential forms
ω=
(z−y)dx+ (x−z)dy+ (y−x)dz
(x−z)
2
+ (y−z)
2
andσ= (z−y)dy∧dz+ (z−x)dx∧dz
onR
3
\L.
(a) Show thatωandσin terms of the coordinates ( ̃x, ̃y, ̃z) = (x−z,y−z,z) are equal to
ω=
− ̃yd ̃x+ ̃xd ̃y
̃x
2
+ ̃y
2
andσ=− ̃yd ̃y∧d ̃z− ̃xd ̃x∧d ̃z.
(b) Employ the coordinates of part (a) and show that both differential forms are closed on
R
3
\Lwith the help of the exterior derivative.
(c) Employ the coordinates of part (a) and compute the differential form
χ=ω∧σ.
(d) Compute the closed contour integral
I
1
=
I
γ
ωwithγ(t) = (x,y,z) = (cos(2t),sin(2t),0) andt∈[0,2π]
and decide on the basis of your result whetherωis exact onR
3
\L.
MAST30031
Methods of Mathematical Physics
Page 2 of 4Semester 2, 2024
(e) Leta∈Rbe fixed. Employ the coordinates of part (a) and find a differential 1-formη
so thatdη=σand compute the surface integral
I
2
=
Z
A
σwithA(r,φ) = ( ̃x, ̃y, ̃z) = (0,rcos(φ)−a,rsin(φ) +a),
wherer∈[0,1],φ∈[0,2π] and the orientation isdr∧dφ= +drdφ,with the help
of the generalised Stokes’ theorem.Do not forget to point out what the
orientation of the boundary ofAis!
Question 3 (6 marks)[Inspired by Question 71 of the Exercise Booklet]
Recall that in polar coordinates, we usually setx=rcosθandy=rsinθ.
(a) Use this to determine the 1-forms drand dθin terms ofxandy.
(b) Specify the maximal subset of the planeR
2
on which these forms are defined.
(c) Give a conclusive argument that dθis closed but not exact and explain why this does not
contradict the notation “dθ” (which suggests exactness).
Question 4 (11 marks)
LetT⊂R
3
be the torus (surface) that is obtained by rotating the circle
S
1
=
(x,0,z)
(x−R)
2
+z
2
=r
2
(where 0< r < R)
about thez-axis.
(a) Translate the vector fieldv(x) =xinto a 2-formωonR
3
.
(b) Compute the integral
R
T
ωin terms of the generalized Stokes Theorem where the orienta-
tion ofTis chosen to be the outward normal. You may use that the volume of the solid
torus is given byV= 2π
2
Rr
2
.
(c) Parametrize the torus in terms of two anglesθandφwhereθ∈[0,2π] describes the angle
of rotation about thez-axis andφ∈[0,2π] describes the angle on the associated circle
away from thexy-plane (withφ= 0 being the outermost point).
(d) Verify that your parametrization satisfies the identity
(
p
x
2
+y
2
−R)
2
+z
2
=r
2
.
(e) Express the 2-formωon the torusTin terms of the coordinatesθandφ.
MAST30031
Methods of Mathematical Physics
Page 3 of 4Semester 2, 2024
Question 5 (18 marks)[Question 76 of the Exercise Booklet]
In this exercise, we study Maxwell’s equations of electromagnetism, namely (ignoring physical
constants!)
divB= 0,
∂B
∂t
=−curlE,divE=ρand
∂E
∂t
= curlB−J,
using the language of differential forms onU⊆R
4
. [You don’t need to know any electro-
magnetism to do this question, though you should marvel at the unbearable coolness of the
translation. We will use the convention thatE=E
x
i+E
y
j+E
z
kandB=B
x
i+B
y
j+B
z
k,
whilst the fourth coordinate ofUist(time).]
(a) Find an antisymmetric 4×4 matrix (F
ij
) such thatthe first twoMaxwell equations
reduce to dF= 0, whereF=
P
4
i,j=1
F
ij
dx
i
∧dx
j
∈Ω
2
(U).
Hint:It is useful to think ofBas described by a 2-form not involvingdtandEas a
1-form multiplied bydt(using the wedge product).
(b) In a vacuum, the scalar fieldρ(the electric charge density) and the vector fieldJ(the
electric current density) both vanish. FindG∈Ω
2
(U) such that the last two Maxwell
equations reduce, in a vacuum, to dG= 0.
(c) Now defineJ∈Ω(U) that allows you to write the general Maxwell equations in differen-
tial form language.
(d) Show that d
2
= 0 implies that electric charge is conserved, meaning that
∂ρ
∂t
+ divJ= 0.
(e) IfU=R
4
, the fact thatF∈Ω
2
(R
4
) is closed means that there existsA∈Ω
1
(R
4
) with
F= dA. WritingA=A
x
dx+A
y
dy+A
z
dz+φdt, expressEandBin terms of
A=A
x
i+A
y
j+A
z
kandφ. Enjoy the fact that the result is not nearly as nice as
F= dA.
End of Assignment — Total Available Marks = 60
MAST30031
Methods of Mathematical Physics
Page 4 of 4Semester 2, 2024
