EQUATIONS OF ELASTIC HYPERSURFACES

3. CALCULUS OF DIFFERENTIAL OPERATORS ON HYPERSURFACES 71
be a first-order differential operator with real valued (variable) matrix coefficients, acting on
vector-valued functions u = (u β ) β in R n and its principal symbol is given by the matrixvalued
function
n∑
σ(P ; ξ) := a j ξ j ξ = { } n
ξ j ∈ j=1 Rn . (3.21)
j=1
Definition 3.10 We say that P is a weakly tangential operator to the hypersurface S , with
unit normal ν, provided
σ(P ; ν) = 0 on the hypersurface S . (3.22)
Next, call P a strongly tangential operator to S provided there exists an extended unit
field N such that
σ(P ; N ) = 0 in an open neighborhood of S in R n . (3.23)
Note that in a strongly tangential operator the derivatives ∂ j can be replaced by the
Gunter’s derivatives D j :
P (∇)u =
n∑
a j ∂ j u + bu =
j=1
n∑
a j D j u + bu = P (D)u , a j , b ∈ C 1 (R m×m ) (3.24)
j=1
Most important tangential differential operators to the hypersurface are for us:
A. The weakly tangential Günter’s derivatives
D j := ∂ j − ν j ∂ ν = ∂ j − ν j
n∑
ν k ∂ k , j = 1, . . . , n ,
introduced in (1.10);
B. The weakly tangential Stokes’s derivatives M jk = ν j ∂ k − ν k ∂ j , introduced in § 2.5.
The Günter’s and Stoke’s derivatives are tangential since the corresponding vector fields
are tangential
k=1
D j := ∂ dj = d j · ∇ , M jk := ∂ mjk = m jk · ∇ ,
d j := π S e j = e j − ν j ν = ν ∧ ( )
n∑
ν ∧ e j = (δ jk − ν j ν k )e k ,
m jk := ν j e k − ν k e j , 〈d j , ν〉 = 0 , 〈m jk , ν〉 = 0 , j, k = 1, . . . , n ,
k=1
(3.25)
where π S is the projection on the tangential space to the surface (cf. (1.11)). Therefore D j
and M jk can be applied to functions which are defined only on the surface S .
The generating vector fields { } n { } n
d j mjk are not frame since they are linearly
j=1 j,k=1
dependent
n∑
ν j (X )d j (X ) ≡ 0 , m jj = 0 , (3.26)
j=1