§ 3. Integration of Differential Forms
215
To finish the proof of theorem 4, it remains to observe that if the borders
A−U and B −V have measure zero,
45 the previous integrals remain invariant
if U and V are replaced by A and B. This indicates that the essential result
is in fact the previous formula, which does not assume anything about the
borders of U and V .
(iv) Stokes’ formula for a p-dimensional path. The definition of the extended integral of a form of degree 2 over a 2-dimensional path σ can be
generalized in an obvious manner: if ω is a form of degree p on an open
subset G of a Cartesian space E and if σ : I
p
−→ G is a a p-dimensional
path (or singular cube) assumed to be of class at least C
1 so that the partial
derivatives of σ extend by continuity to the border of I
p , then, by definition,
set
σ
ω =
I p
ω [σ(t); D 1 σ(t), . . . , D p σ(t)] dt ,
(10.27)
where t = (t
1 , . . . , t
p ) and where dt = dt
1 . . . dt
p is the usual Lebesgue measure. This obviously amounts to setting
ω ◦ σ = r(t)dt
1
∧ . . . ∧ dt
p
and
σ
ω =
I p
r(t)dt
as in degree 2. If σ is replaced by σ ◦ ϕ, where ϕ : I
p
−→ I
p is a diffeomorphism, then, by the associativity formula (8.19), ω ◦ σ = is replaced
by
ω ◦ (σ ◦ ϕ) = (ω ◦ σ) ◦ ϕ = ◦ ϕ .
So r(s) is replaced by r[ϕ(t)]J ϕ (t). Formula (2), seen to be equivalent to
Theorem 4, then shows that
σ◦ϕ
ω = sgn(ϕ)
σ
ω ,
(10.28)
where sgn(ϕ) is the, necessarily constant, sign of the Jacobian of ϕ.
This formula resembles the definition of the integral of a form ω along a
path σ such as the integral over the cube of its inverse image under σ, but
they should not be confused : (28) is a theorem and not a definition. More
generally, consider the open subsets U and V of two Cartesian spaces, a C
1
map f : U −→ V and a path p-dimensional σ in U . This gives a path f ◦ σ
in V . Then
f ◦σ
ω =
σ
ω ◦ f
(10.29)
45 By lemma 4, this is always the case when ϕ can be extended to a C
1 function
defined on an open set containing A.