1. Let M m be a compact oriented manifold, ω ∈ Ωp−1 (M ), θ ∈ Ωm−p (M ). Prove the
following partial integration formula:
Z
Z
Z
p
dω ∧ θ =
ω ∧ θ + (−1)
ω ∧ dθ.
M
∂M
M
2. Let M ⊂ Rn be a bounded open set with smooth boundary S = ∂M . For p ∈ S let
N (p) be the unit vector, which is orthogonal to τp (S) and points outward.
Show that the volume form ωS on S is characterized by the condition that if v1 , . . . , vn−1
is an oriented basis of τp (S), then ωS (p)(v1 , . . . , vn−1 ) is the
of the parallelepiped
P volume
∂
n
in R defined by N (p), v1 , . . . , vn−1 . Conclude that if A = i Ai ∂xi is a vector field on Rn ,
then the restriction of
ω=
n
X
ˆ i ∧ . . . ∧ dxn
(−1)i−1 Ai dx1 ∧ . . . ∧ dx
i=1
P
to S is equal to (A, N )ωS , where (A, N )(p) = i Ai (p)Ni (p).
Then prove the divergence theorem:
Z
Z
(A, N )ωS =
div A ωM ,
S
where ωM = dx1 ∧ . . . ∧ dxn and div A =
M
∂Ai
i ∂xi
P
is the divergence of the vector field A.
3. Let S ⊂ R3 be an oriented surface with boundary L = ∂S. Let N (p) be the unit
vector, which is orthogonal to τp (S) and defines the orientation of S, that is, v1 , v2 is an
oriented basis of τp (S) if N (p), v1 , v2 is an oriented basis of R3 . For p ∈ L let `(p) ∈ τp (L) be
the unit tangent vector pointing in the positive direction. Denote
P by ωS and ωL the volume
forms of S and L, respectively. By considering the form ω = i Ai dxi prove the classical
Stokes theorem:
Z
Z
(A, `)ωL = (rot A, N )ωS ,
L
where
S
∂A3 ∂A2 ∂A1 ∂A3 ∂A2 ∂A1
rot A =
−
,
−
,
−
∂x2
∂x3 ∂x3
∂x1 ∂x1
∂x2
P
is the rotation of the vector field A = i Ai ∂x∂ i , sometimes also denoted by curl A.