PROBLEM SET IX
DUE FRIDAY,  MAY
Exercise . Fix vectors v1 , v2 , . . . , vn−1 ∈ Rn . Show that there exists a unique vector x ∈ Rn such that for any
w ∈ Rn , one has
x · w = det(v1 , v2 , . . . , vn−1 , w).
In this situation, we call x the cross product of v1 , v2 , . . . , vn−1 , and we write
x = v1 × v2 × · · · × vn−1 .
Note that the cross product is a map (Rn )n−1 .
than n − 1 vectors!
Rn ; it does not make sense to speak of the cross product of fewer
Exercise . Show that the cross product is an alternating multilinear map (Rn )n−1 .
vectors v1 , v2 , . . . , vn−1 ∈ Rn , one has
√
|v1 × v2 × · · · × vn−1 | = det M,
Rn , and show that, for any
where M is the (n − 1) × (n − 1) matrix whose (i, j)-th entry is vi · vj .
Definition. e divergence of a vector field F = (f1 , f2 , . . . , fn ) on an open subset U ⊂ Rn is the function ∇ ·
F : U . R given by the formula
n
∑
∂fi
.
∇ · F :=
∂xi
i=1
When n = 3, we can also speak of the curl of a vector field F = (f1 , f2 , f3 ); this is a new vector field
(
)
∂f3
∂f ∂f
∂f ∂f
∂f
∇×F=
− 2, 1 − 3, 2 − 1
∂x2 ∂x3 ∂x3 ∂x1 ∂x1 ∂x2
on U.
Finally, of course, the gradient of a function g : U . R is the vector field
(
)
∂g ∂g
∂g
∇g :=
,
,··· ,
.
∂x1 ∂x2
∂xn
Exercise . Explain the intuition behind the notation we have used for the divergence and the curl. By using what
you have learned about the cross product, can you define a generalization of the curl to higher dimensions?
Now for any vector field F = (f1 , f2 , f3 ) on an open subset U ⊂ R3 , write
ω1F := f1 dx + f2 dy + f3 dz
ω2F := f3 dx ∧ dy − f2 dx ∧ dz + f1 dy ∧ dz
R on an open subset U ⊂ R3 , one has
Exercise . Prove that for any function g : U .
dg = ω1∇g ,
and show that for any vector field F on U, one has
dω1F = ω2∇×F
and dω2F = (∇ · F)dx ∧ dy ∧ dz.

Exercise . Show that, for any function g : U .
R on an open subset U ⊂ R3 , one has
∇ × (∇g) = 0,
and for any vector field F on U,
∇ · (∇ × F) = 0.
Conversely, show that if U is contractible, then for any vector field F on U such that ∇ × F = 0, there exists a
function g : U . R such that F = ∇g, and for any vector field F on U such that ∇ · F = 0, there exists a vector
field G on U such that F = ∇ × G.
Exercise . Suppose M ⊂ R3 a compact 3-manifold. For any vector field F on M, express the integral
∫
(∇ · F) dx ∧ dy ∧ dz
M
as an integral over ∂M.
Exercise . Suppose N ⊂ R3 an oriented, compact 2-manifold. For any vector field F on an open subset of R3
containing N, express the integral
∫
N
ω2∇×F
as an integral over ∂N.
