1. No category

Math 52H: Homework N4
Due to Friday, February 6
1. Consider in R2n differential forms
ω = dx1 ∧ dx2 ∧ · · · ∧ dxn and θ =
2n
X
(−1)j−1 xj dxn+1 ∧ · · · ∧ dxj−1 ∧ dxj+1 ∧ · · · ∧ dx2n .
n+1
Prove that dΩ = 0, where the (2n − 1)-form Ω is defined by the formula
ω∧θ
.
Ω= P
n
n
( xi xi+n )
1
2. In Rn \ 0 for every integer k consider a differential (n − 1)-form
n
i
1 X
θk := k
(−1)i−1 xi dx1 ∧ · · · ∧ ∨ ∧ . . . dxn ,
r 1
v
u n
uX
where r = t
x2i .
1
Find all values of k for which the form dθk = 0.
3. Consider a differential 1-form ω = F1 dx + F2 dy + F3 dz in R3 which satisfies dω = 0.
Suppose that each function Fk , k = 1, 2, 3, satisfies the homogeneity equation
Fk (tx, ty, tz) = tFk (x, y, z),
for any t ∈ R. Prove that ω = df , where
f (x, y, z) =
1
(xF1 (x, y, z) + yF2 (x, y, z) + zF3 (x, y, z)) .
2
1
4. Let Rn is endowed with the standard dot-product and orientation. Consider an operator
∂ = (−1)k ?−1 d? : Ωk (Rn ) → Ωk−1 (Rn ).1 Denote ∆ := (∂ + d)2 = ∂d + d∂ (this is called
Laplace-de Rham operator). Verify that
∂ can be equivalently defined by the formula ∂ = (−1)nk+n+1 ? d? ;
∆ is an operator Ωk (Rn ) → Ωk (Rn ) ;
and find explicit formulas in Cartesian coordinates for ∆α for the cases when n = 2 and
k = 0, 1, 2.
Each problem is 10 points.
1
For a differential k-form we define the star operator by the formula (?α)x = ?(αx ).
2