School of Mathematics and Statistics
MAST30031 Methods of Mathematical Physics, Semester 2 2023
Written assignment 3 and Cover Sheet
Student Name Student Number
Submit your assignment solutions together with this coversheet via the MAST30031 Gradescope
website before Friday 13th October (9am AET) sharp. No extensions will be granted! Only
exemptions for suitably justified reasons can be granted.
• This assignment is worth 10% of your final MAST30031 mark.
• Assignments must be either neatly handwritten or can be written with LaTeX.
• Full working must be shown in your solutions.
•Marks will be deducted in every question for incomplete working, insufficient justification of steps and
incorrect mathematical notation.
•Youmust usemethods taught inMAST30031Methods ofMathematical Physics to solve the assignment
questions.
• All tasks are mandatory for everyone!
• There are in total 40 points to achieve.
• Begin your answer for each question on a new page!
Please, turn the page for the other questions!
Page 1 of 7
1. Summary 10 points.
Write a summary of the third part of the lectures called “Differential Forms”. The summary should be
between two and three A4 pages! To make it simple, pick ten out of the sixteen topics below and briefly
describe those in a concise way. Use the space on the following three pages.
• differential 0-forms and 1-forms as vector spaces
• differential 1-forms as linear maps
• 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
• Levi-Cevita symbol and the determinant
• Hodge star operator
• the Laplacian for differential forms
• de Rahm cohomologies
•Maxwell’s equations
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Page 3 of 7
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Page 5 of 7
2. Simple Question 10 points.
Compute the following contour integrals
(a) Compute the 1-dim contour integral
I1 =
∫
γ
ω
for the differential 1-form
ω = ydx + xdy − xydz
and the 1-dim contour
γ (t) = (x,y, z) = (et , t, t2).
with t ∈] − ∞, 0].
(b) Compute the 2-dim contour integral
I2 =
∫
S
σ
for the differential 2-form
σ = ydx ∧ dy + xdy ∧ dz − yxdx ∧ dz
and the 2-dim contour
S(s, t) = (x,y, z) = (et , t + s, t2 + s2).
with t ∈ [−∞, 0] and s ∈ [0, 1] and the orientation ds ∧ dt = +dsdt .
3. Moderate Question 10 points.
Consider the differential forms
ω1 =
zx√
x2 + y2[(1 − √x2 + y2)2 + z2]dx + zy√x2 + y2[(1 − √x2 + y2)2 + z2]dy + (1 −
√
x2 + y2)
(1 − √x2 + y2)2 + z2dz,
ω2 =
x
x2 + y2
dy − y
x2 + y2
dx .
(a) Change the coordinates to those on the torus
T =
{
(x,y, z) ∈ ’3 :
(
1 −
√
x2 + y2
)2
+ z2 =
1
4
}
,
which are
(x,y, z) =
((
1 − 1
2
cos(ϑ )
)
cos(φ),
(
1 − 1
2
cos(ϑ )
)
sin(φ), 1
2
sin(ϑ )
)
with ϑ ,φ ∈ ’, and show that the two differential forms become
ω1 = dϑ and ω2 = dφ.
Hint: this computation can be messy if you do not do it in the proper order. First, simplify all coefficient
functions and then compute the differentials. Once this is done put everything together.
(b) Compute the wedge product σ = ω1 ∧ ω2 in these new coordinates.
(c) Show that ω1, ω2, and σ are closed on the torus T .
Hint: you can use the drastically simplified form.
(d) Explain why the integrals
I1 =
1
2pi
∮
γ
ω1, I2 =
1
2pi
∮
γ
ω2
are integers for every closed curve γ : [0, 1] → T . Deduce from this that ω1 and ω2 are inexact on the
torus T .
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Page 6 of 7
4. Challenge Question 10 points.
LetU ⊂ ’N and we choose the orientation dx1∧· · ·∧dxN = +dx1 · · ·dxN . Consider real valued differential
forms ω,σ ∈ Ωk (U ) with k ≤ N real valued differential k-form and a general Riemannian metric following
from the length element
ds2 =
N∑
a,b=1
dxaдabdx
b .
Then, we define
〈ω,σ 〉 =
∫
U
ω ∧ (∗σ ).
(a) Explain why it is
dxbs (1) ∧ . . . ∧ dxbs (k ) = ϵs(1)...s(k )dxb1 ∧ . . . ∧ dxbk ,
for any permutation s : {1, . . . ,k} → {1, . . . ,k} (meaning s ∈ “k the symmetric group), where
ϵs(1)...s(k ) is the Levi-Cevita symbol. Why is then
dxb1 ∧ . . . . . . ∧ dxbN = ϵb1 ...bNdx1 ∧ . . . ∧ dxN
true for all b1, . . . ,bN ∈ {1, . . . ,N }?
(b) Show that
N∑
bk+1, ...,bN =1
(ϵb1 ...bN )2 = (N − k)!
for all fixed b1, . . . ,bk ∈ {1, . . . ,N } satisfying bl , bj for all l , j for all l, j = 1, . . .k.
Hint: the number of permutations in the symmetric group “l is l!.
(c) With the help of (a) and (b), show that
〈ω,σ 〉 = k!
∫
U
©­«
N∑
a1, ...,ak ,b1, ...,bk=1
ωa1 ...akд
a1b1 · · ·дakbkσb1 ...bk
ª®¬
√
det(д)dx1 · · ·dxN
for any two square integrable differential forms
ω =
N∑
a1, ...,ak=1
ωa1 ...akdx
a1 ∧ · · · ∧ dxak and σ =
N∑
b1, ...,bk=1
σb1 ...bkdx
b1 ∧ · · · ∧ dxbk .
Recall that the coefficient functions are skew-symmetric in their indices!
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