Student Name
School of Mathematics and Statistics
MAST30031 Methods of Mathematical Physics, Semester 2 2023 Written assignment 3 and Cover Sheet
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.
• You must use methods taught in MAST30031 Methods of Mathematical 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!
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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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Programming Help
2. Simple Question 10 points.
Compute the following contour integrals
(a) Compute the 1-dim contour integral
for the differential 1-form
and the 1-dim contour
witht ∈]−∞,0].
(b) Compute the 2-dim contour integral
ω = ydx + xdy − xydz γ(t) = (x,y,z) = (et,t,t2).
for the differential 2-form
and the 2-dim contour
witht ∈[−∞,0]ands ∈[0,1]andtheorientationds∧dt =+dsdt.
3. Moderate Question 10 points. Consider the differential forms
σ = ydx ∧ dy + xdy ∧ dz − yxdx ∧ dz S(s,t) = (x,y,z) = (et,t +s,t2 +s2).
(1−px2 +y2) ω1=p2 2 p2 22 2dx+p2 2 p2 22 2dy+ p2 22 2dz,
x+y[(1− x+y)+z] x+y[(1− x+y)+z] (1− x+y)+z
ω2= x dy− y dx. x2 +y2 x2 +y2
(a) Change the coordinates to those on the torus
T= (x,y,z)∈3: 1− x2+y2 +z2=4 ,
1 1 1
(x,y,z) = 1 − 2 cos(θ) cos(φ), 1 − 2 cos(θ) sin(φ), 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 ∮ ω1, I2 = 1 ∮ ω2 2π γ 2π γ
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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4. Challenge Question 10 points.
LetU ⊂N andwechoosetheorientationdx1∧···∧dxN =+dx1···dxN.Considerrealvalueddifferential forms ω, σ ∈ k (U ) with k ≤ N real valued differential k-form and a general Riemannian metric following from the length element
ds2 = Õ dxaдabdxb.
εs(1)…s(k) is the Levi-Cevita symbol. Why is then
dxb1 ∧……∧dxbN =εb1…bNdx1 ∧…∧dxN
trueforallb1,…,bN ∈ {1,…,N}? (b) Show that
(εb1…bN )2 = (N −k)! bk+1,…,bN =1
forallfixedb1,…,bk ∈{1,…,N}satisfyingbl ,bj foralll,jforalll,j=1,…k. Hint: the number of permutations in the symmetric group l is l!.
Then, we define
(a) Explain why it is
for any permutation s : {1,…,k} → {1,…,k} (meaning s ∈ k the symmetric group), where
(c) With the help of (a) and (b), show that
∫ ÕN ⟨ω,σ⟩=k! ©
abab p1N ωa …a д 1 1 ···д k kσb …b a® det(д)dx ···dx
dxbs(1) ∧…∧dxbs(k) =εs(1)…s(k)dxb1 ∧…∧dxbk,
ω ∧ (∗σ ).
1k 1k «1k1k ¬
U a ,…,a ,b ,…,b =1
for any two square integrable differential forms
ω = Õ ωa1…akdxa1 ∧···∧dxak and σ = Õ σb1…bkdxb1 ∧···∧dxbk .
a1,…,ak =1 b1,…,bk =1
Recall that the coefficient functions are skew-symmetric in their indices!
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