EQUATIONS OF ELASTIC HYPERSURFACES

44 SHELLS
Definition 2.49 Let { S , β S
}
be a smooth hypersurface with a boundary Γ = ∂M , oriented
by the non-vanishing differential form of maximal order β S ∈ Λ n−1 (S ).
The natural orientation of the boundary Γ is then specified by the restriction β Γ = β S
∣
∣Γ
which is a form of maximal order β Γ ∈ Λ n−2 (Γ).
The next Lemma 2.50 provides an useful and interesting example of restriction of the
differential form to hypersurface and to it’s boundary.
Lemma 2.50 Let Θ : Ω → S be a smooth hypersurface in R n with a smooth boundary
Γ := ∂S , while dS and ds designate the respective volume elements on S and on Γ.
Let ν(X ) = ( ν 1 (X ), . . . , ν n (X ) ) ⊤
be the outer unit normal vector to S at X ∈ S and
ν Γ (s) = ( ν 1 Γ (s), . . . , νn Γ (s)) ⊤
-the unit tangential vector to S at the boundary point s ∈ Γ,
which is outward (unit) normal vector to the boundary S . Then
∣
ν j dS = β ∣S j , (2.174)
[
νj νΓ k − ν ] ∣
kν j Γ ds = βjk∣Γ , (2.175)
where
β j := ∣ ∣ dx1 ∧ . . . ∧ ̂dx j ∧ . . . ∧ dx n
∣ ∣ = (−1) j−1 dx 1 ∧ . . . ∧ ̂dx j ∧ . . . ∧ dx n ,
β jk := ∣ ∣dx 1 ∧ . . . ∧ ̂dx j ∧ . . . ∧ ̂dx k ∧ . . . ∧ dx n
∣ ∣
= (−1) j+k−1 dx 1 ∧ . . . ∧ ̂dx j ∧ . . . ∧ ̂dx k ∧ . . . ∧ dx n
and the “hat” symbol indicates omission.
Proof: According to Lemma 2.29 the differential form β j generates the volume element
of the hyperplane x j = const in R n . On the other hand, being a positive ∣ (n − 1)-form,
the restriction to the surface S is a maximal form and, due to (2.140), β ∣S j = g j dS for
some function g j (X ). To find the function g j (X
∣ ) it suffices to consider the restriction to the
hyperplane β ∣xk j = β =const j, where it defines the volume element. The projection of the
volume element dS of S to the hyperplane x j = const is ν j dS = cos θ j dS and, therefore,
g j (X ) ≡ ν j (X ).
∣
By applying the above argument on projection to the (n − 2)-form β ∣Γ jk = g jk ds which
is maximal on Γ, we get similarly g jk = ν j νΓ k − ν kν j Γ
and formula (2.175) follows.
Theorem 2.51 Let β be a compactly supported (n − 2)-form with coefficients in the class
C 1 on an oriented n-dimensional 2-smooth manifold S with boundary Γ oriented naturally.
The following Stoke’s formula is valid:
∫ ∫
dβ := β . (2.176)
S
Γ