EQUATIONS OF ELASTIC HYPERSURFACES

2. OUTLINE OF DIFFERENTIAL GEOMETRY 41
To this end note that
Θ ◦ F = Y and Y # β S = F # Θ # β S . (2.157)
In fact, it suffices to check the second equality for the basic differential 1-form dx j (cf.
(2.149)). Then Θ # dx j = ∑ ( ∂Θ/∂x k
)
dxk and further
F # Θ # dx j = ∑ k,m
∂F k
∂x m
∂Θ j
∂x k
dx m ,
Y # dx j = ∑ k
∂Y j
∂x m
dx m . (2.158)
The identity of the forms is a consequence of the chain rule:
DY = DΘDF
i.e.,
∂Y j
∂x k
= ∑ k,m
∂Θ j
∂x k
∂F k
∂x m
.
Now the claimed equality (2.156) follows by successive application of equalities (2.157) and
(2.153): ∫ ∫
∫
Y # β S := F # Θ # β S = Θ # β S .
V
V
Next we introduce the exterior product of antisymmetric (exterior) k-forms.
Definition 2.43 The exterior product of an exterior form β ∈ Λ k given by formulae (2.137)
and of exterior form δ ∈ Λ m
U
δ(X ) = ∑ j
b j1 ...j m (X ) dx j1 ∧ . . . ∧ dx jm , (2.159)
equals
β ∧ δ =
n∑ n∑
a j1···j k
b l1···l m
dx j1 ∧ · · · ∧ dx jm ∧ dx l1 ∧ · · · ∧ dx lm . (2.160)
j 1 ,··· ,j k =1 l 1 ,··· ,l m =1
Note that to the antisymmetric products in the sum (2.160) apply the rules postulated in
(2.138) and Corollary 2.36. In particular, if j p = l q for some p = 1, . . . , k and q = 1, . . . , m,
then the corresponding summand in (2.160) eliminates. Moreover, if k +m > 0 then β ∧δ =
0 while for k + m = n the product consists of a single summand (cf. (2.140)).
Lemma 2.44 The exterior product of antisymmetric forms defines a mapping
which is:
i. bilinear
Λ k × Λ m → Λ k+m (2.161)
β ∧ (cδ + dγ) = cβ ∧ cδ + dβ ∧ γ , (cδ + dγ) ∧ β = cδ ∧ β + dγ ∧ β ; (2.162)