EQUATIONS OF ELASTIC HYPERSURFACES

4 SHELLS
The form is correctly defined because the differential operator ∂ V is tangential and can be
applied to a function f defined on the surface S only.
Then, for a given f, there exists a vector field ∇ S f ∈ TS such that
ω f (V ) := 〈∇ S f, V 〉 for all V ∈ V (X) , (1.3)
which is, classical differential geometry, the definition of surface Gradient of a function
f ∈ C 1 (S ) and maps
∇ S : C ∞ (S ) → TS . (1.4)
The surface divergence
Div S : V (S ) → C ∞ (S ) (1.5)
of a smooth tangential vector field is, by the definition, the dual operator with the opposite
sign
〈Div S V, f〉 := −〈V, ∇ S f〉 , ∀ V ∈ TS , ∀ f ∈ C 1 (S ) . (1.6)
Expressing these operators in intrinsic parameters of the surface S (tangential vector fields,
Metric tensor etc.) one obtains (cf. [Ca1, Ci4, Ku1, Ta2]):
Div S U := [ det G S
] −1/2
n∑
j=1
{ [det ] } 1/2U ∑
j
∂ j GS = n
∇ S f = ∑ j,k (gjk ∂ j f) ∂ k .
j=1
U j
;j = Tr ∇ S U ,
(1.7)
where U j
;k denotes the componentwise covariant derivative, while ∇ S U is the tensor field
defined by the covariant derivative (∇ S U)V := ∂U S V (see § 2.7 and [Ta2, Ch. 2, § 3]).
Their composition is the Laplace-Beltrami operator
∆ S f := Div S ∇ S f = [ ] −1/2
∑n−1
det G S
which is self-adjoint due to (1.5):
j,k=1
∂ j
{g jk[ ] }
1/2∂k
det G S f , f ∈ C 2 (S ) ,
(1.8)
∆ ∗ S = ( ∇ S Div S
) ∗
=
(
DivS
) ∗ (
∇S
) ∗
= ∇S Div S = ∆ S . (1.9)
The intrinsic parameters enable generalization to arbitrary manifolds, not necessarily immersed
in the Euclidean space R n .
On the other hand it is sometimes more convenient to record these operators in cartesian
coordinates. For example, it is easy to ascertain that the surface gradient is nothing but the
collection of the weakly tangential Günter’s (covariant) derivatives
∇ S = D S := (D 1 , . . . , D n ) ⊤ , D j := ∂ j − ν j (X )∂ ν = ∂ dj , (1.10)
where ∂ ν := ∑ n
j=1 ν j∂ j denotes the normal derivative (cf. [Gu1], [KGBB1], [Du1]). Here,
for each 1 ≤ j ≤ n, the first-order differential operator D j = ∂ dj is the directional derivative
along the tangential vector d j := π S e j . The operator (the matrix)
π S : R n → TS , π S (t) = I − ν(t)ν ⊤ (t) = [ δ jk − ν j (t)ν k (t) ] n×n , t ∈ S (1.11)