EQUATIONS OF ELASTIC HYPERSURFACES

2. OUTLINE OF DIFFERENTIAL GEOMETRY 29
Therefore an integral on the surface is defined by the equality (we resume temporarily the
indexation Θ j )
∫
=
S
f(X )dS :=
M∑
∫
j=1
M∑
∫
j=1
S j
ψ j (X )f(X )|d 1 Θ j ∧ . . . ∧ d n−1 Θ j |
ω j
ψ j (Θ j (y))f(Θ j (y)) √ det G S (y))dy , (2.102)
where {ψ j } M j=1 is a partition of unity subordinated to the chosen covering {S j } M j=1 of S .
To prove the correctness of the definition we have to check the independence of the integral
in the right-hand side in (2.102) from the variable transformation which preserves the orientation.
It is sufficient to check this for one chart M = 1 and ω j = ω, S j = S , Θ = Θ.
Let
y = y(z) : ω → W ⊂ R n−1 (2.103)
be a diffeomorphism which preserves the orientation and κ(z) := Θ(y(z)). Then
∑n−1
∂ k κ(z) = (∂ m Θ)(y(z))∂ m y(z) , k = 1, . . . , n − 1 .
m=1
Due to Lemma 2.25.g and the equality (2.92)
det G S (z)=G ((∂ 1 Θ)(y(z)), . . . , (∂ n−1 Θ)(y(z))) = |(∂ 1 Θ)(y(z)) ∧ . . . ∧ (∂ n−1 Θ)(y(z))|
=[det[∂ j y k (z)] n−1×n−1 ] −1 G (∂ 1 κ(z), . . . , ∂ n−1 κ(z)) , (2.104)
where J(z) := det[∂ j y k (z)] n−1×n−1 is the Jakobi determinant of the transformation. Changing
the variable y = y(z) in the integral (2.102) and applying the well known formula
dy(z) = det[∂ j y k (z)] n−1×n−1 dz we obtain
∫
ψ ◦ Θ(y)f ◦ Θ(y) √ det G S (y))dy
ω
∫
= ψ ◦ κ(z)f ◦ κ(z) √ (z)dz ,
W
which proves that the integrals in the right-hand side in (2.102) are independent of the variable
transformation.
In particular, the surface area A (S ) of the hypersurface S is given by the formula
(2.102) with f ◦ Θ j = 1 (we resume, again temporarily, the indexation Θ j ):
∫
A (S ) =
S
dS =
=
M∑
∫
ψ j ◦ Θ|d 1 Θ j ∧ . . . ∧ d n−1 Θ j |
S j
M∑
∫
ψ j (Θ j (y)) √ det G S (y)dy . (2.105)
ω j
j=1
j=1