PROBLEM SET IX
DUE FRIDAY,  MAY
De nition. e divergence of a vector eld 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 eld F = (f1 , f2 , f3 );
this is a new vector eld
(
)
∂f3
∂f2 ∂f1
∂f3 ∂f2
∂f1
∇×F=
−
,
−
,
−
∂x2 ∂x3 ∂x3 ∂x1 ∂x1 ∂x2
on U.
Finally, of course, the gradient of a function g : U . R is the vector eld
(
)
∂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 de ne a generalization of the curl to higher dimensions?
Now for any vector eld F = (f1 , f2 , f3 ) on an open subset U ⊂ R3 , write
ωF1 := f1 dx + f2 dy + f3 dz
ωF2 := f3 dx ∧ dy − f2 dx ∧ dz + f1 dy ∧ dz
Exercise . Prove that for any function g : U .
one has
1
,
dg = ω∇g
R on an open subset U ⊂ R3 ,
and show that for any vector eld F on U, one has
2
dωF1 = ω∇×F
and
dωF2 = (∇ · F)dx ∧ dy ∧ dz.
Exercise . Show that, for any function g : U . R on an open subset U ⊂ R3 ,
one has
∇ × (∇g) = 0,


DUE FRIDAY,  MAY
and for any vector eld F on U,
∇ · (∇ × F) = 0.
Conversely, show that if U is contractible, then for any vector eld F on U such
that ∇ × F = 0, there exists a function g : U . R such that F = ∇g, and for
any vector eld F on U such that ∇ · F = 0, there exists a vector eld G on U
such that F = ∇ × G.
Exercise . Use the Poincaré lemma to show that there exists a smooth function g : R . R such that
(sin x + x2 exp(g(x)) − 1)g′ (x) + g(x) cos x + 2x exp(g(x)) = 0.
De nition. For any real numbers 0 ≤ r ≤ s, let T2 (r, s) ⊂ R3 denote the torus
{
}
2
2
2
2
2 2
2 2
2
2
3 T (r, s) := (x, y, z) ∈ R (x + y + z + s − r ) = 4s (x + y ) .
∪
Let us write ST(r, s) for the union ρ∈[0,r] T2 (ρ, s).
Exercise . What is the volume of ST(r, s)?
Exercise . What is the surface area of T2 (r, s)?
Exercise . Consider the following vector eld on R3 − {(0, 0, 0)}:
1
ϕ(x, y, z) = 2
(−y, x, (z − 1/4)(z + 1/4) − x2 − y2 ).
x + y2 + z2
Is this vector eld a gradient vector eld?