EQUATIONS OF ELASTIC HYPERSURFACES

2. OUTLINE OF DIFFERENTIAL GEOMETRY 43
for a smooth mapping F : M → S . In fact, due to (2.168) we get
dF # β = ∑ j,l
+ ∑ j,m
∂ (
αj ◦ F (x) ) dx l ∧ ( ) ( )
F # dx j1 ∧ · · · ∧ F# dx jk
∂x l
(−1) m−1 α j
(
F (x)
)(
F# dx j1
)
∧ · · · ∧ d
(
F# dx jm
)
∧ · · · ∧
(
F# dx jk
)
= ∑ j,l
∂ (
αj ◦ F (x) ) dx l ∧ ( ) ( )
F # dx j1 ∧ · · · ∧ F# dx jk
∂x l
(2.171)
because d ( F # dx m
)
= d
(
dzm
)
= 0 where zm (y) := x m
(
F (y)
)
(cf. (2.168)) and the second
sum in (2.171) vanishes.
Meanwhile,
F # dβ = ∑ j,m
∂α j
∂x m
(
F (x)
)
∧
(
F# dx m
)
∧
(
F# dx j1
)
∧ · · · ∧
(
F# dx jk
)
, (2.172)
so (2.169) follows from (2.170) and the identity
∑
m
∂
∂x m
(
αj ◦ F (x) ) dx m = ∑ m
∂α j
∂x m
(
F (x)
)
F# dx m , (2.173)
which in turn is a consequence of the chain rule.
Definition 2.46 If dβ = 0 we say that β is closed.
If β = dγ for some γ ∈ Λ k−1 (S ), we say that β is exact.
From (2.168) there follows.
Corollary 2.47 Each exact form is closed.
The inverse assertion is globally false, but is valid locally (cf. [Shu1, § 7.24] for the
local version).
Next we discuss a basic result of the theory of differential forms-the Stoke’s formula,
which is a generalization of the classical formula (2.129) and (2.130)). To this end we need
to clear what is a natural orientation of the boundary of a manifold.
Remark 2.48 Having a differential form β S ∈ Λ k (S ) on a k-smooth hypersurface S ⊂
R n with the boundary Γ = ∂S , we consider a restriction β Γ := β ∣ ∂M
. Since β S is a multilinear
functional on vector fields Θ 1 , . . . , Θ k ∈ V (S ), by restricting it to Γ we eliminate
those parts (or summands) in β S which are orthogonal to all tangential vector fields to the
boundary Θ ∈ TΓ.
For example, all those summands in the differential form β := ∑ j
α j dx j1 ∧ · · · ∧ dx jk
on R n − = R − × R n−1 which contain dx 1 will eliminate after restriction to ∂R n − = R n−1 .