§ 3. Integration of Differential Forms
217
The Green-Riemann formula can then be written
K
dω =
ε(F )
F
ω ,
(10.33)
where, as mentioned above, the extended integrals of ω over the faces F of
K = I
p+1 are defined by using the parametric representations (31), which
amounts to regarding these faces as p-dimensional singular cubes.
To go from here to a formula of Stokes type for a form ω of degree p and
an arbitrary path of dimension p+1 in an open subset G of a Cartesian space,
we need to define what will be meant by the extended integral of ω over ∂σ ;
denoting the parametric representation (31) of the face F by ϕ F , it will be
the expression
∂σ
ω =
ε(F )
σ◦ϕ F
ω =
ε(F )
ϕ F
ω ◦ σ .
Using formula (33) for the cube, Stokes’ formula for σ,
∂σ
ω =
σ
dω
then becomes trivial.
The presence of the signs ε(F ) will become clear in n
◦ 16 where a slightly
different version of Stokes’ formula will be proved; as in dimension 1 or 2,
it corresponds to the necessity of choosing an “ orientation ” for each face F
of K.
Exercise. Let σ 0 , σ 1 : I
p
−→ G be two p-dimensional paths in an open
subset G of a Cartesian space, coinciding on the border of I
p (the analogue
of two one-dimensional paths with the same endpoints). They will be said to
be fixed-border-homotopic if there is a path σ : I × I
p
−→ G satisfying the
following conditions : (i) σ(0, t) = σ 0 (t), σ(1, t) = σ 1 (t) for all t ∈ I
p , (ii) for
every point t in the border of I
p , σ(s, t) is independent of s. Show that, if ω
is a closed form of degree p on G, then the extended integrals of ω over σ 0
and σ 1 are equal. Similarly, generalize the invariance under homotopy of the
integral over a closed path.