216
IX – Multivariate Differential and Integral Calculus
for any differential form ω of degree p on V , which is even more like (28). But
(29) means that integrals over the cube of ω ◦ (f ◦ σ) and (ω ◦ f ) ◦ σ are equal;
these two forms being identical by (8.19), there is nothing to show. Relation
(29) is, therefore, almost a tautology or, equivalently, a direct consequence of
the multivariate chain rule formula (2.13).
To take an example, let us return to the computations of n
◦ 9, (ii) related
to the effect of a homotopy σ on the integral of a form ω of degree 1 along a
path ; writing the final formula as
∂σ
ω =
σ
dω ,
(10.30)
it reduces to Gauss’ formula for the square I
2 .
An analogue of the Green-Riemann formula for the cube exists in arbitrary dimension; using Cauchy’s simple calculation, the proof is the same as
in dimension 2. The only difficulty concerns the definition of the extended
integral of a form ω of degree p over ∂K for K = I
p+1 ; it is the sum of
extended integrals over the faces of the cube, but the signs + and − placed
before these integrals need to be determined. However, denoting the canonical
coordinates of R
p+1 by t
0 , . . . , t
p gives formulas of type
ω =
p i (t)dt
0
∧ . . .
dt i . . . ∧ dt
p ,
dω =
(−1)
i D i p i (t)dt
0
∧ . . . ∧ dt
p ,
so that the integral of dω is the sum of the integrals over K of the functions
(−1)
i D i p i (t). To calculate them, first integrate with respect to the corresponding t
i s. This gives (FT)
(−1)
i
I p p 1
t
0 , . . . , 1, . . . , t
p
dt
0 . . . dt
i . . . dt
p +
+(−1)
i+1
I p p 1
t
0 , . . . , 0, . . . , t
p
dt
0 . . . dt
i . . . dt
p
involving the extended integrals of ω over the faces of the cube I
p+1 . More
precisely, write F
+
i for the face t i = 1 of the cube and F
−
i for the face t i = 0,
and define the extended integrals of ω over these faces by using the parametric
representation
ϕ
+
i :
t 0 , . . . ,
t i , . . . , t p
−→ (t 0 , . . . , 1, . . . , t p )
(10.31)
in the first case and the the analogous formula in the second. If F is a face
of the cube, set
ε(F ) =
(−1)
i
if F = F
+
i ,
(−1)
i+1 if F = F
−
i .
(10.32)
IX – Multivariate Differential and Integral Calculus
for any differential form ω of degree p on V , which is even more like (28). But
(29) means that integrals over the cube of ω ◦ (f ◦ σ) and (ω ◦ f ) ◦ σ are equal;
these two forms being identical by (8.19), there is nothing to show. Relation
(29) is, therefore, almost a tautology or, equivalently, a direct consequence of
the multivariate chain rule formula (2.13).
To take an example, let us return to the computations of n
◦ 9, (ii) related
to the effect of a homotopy σ on the integral of a form ω of degree 1 along a
path ; writing the final formula as
∂σ
ω =
σ
dω ,
(10.30)
it reduces to Gauss’ formula for the square I
2 .
An analogue of the Green-Riemann formula for the cube exists in arbitrary dimension; using Cauchy’s simple calculation, the proof is the same as
in dimension 2. The only difficulty concerns the definition of the extended
integral of a form ω of degree p over ∂K for K = I
p+1 ; it is the sum of
extended integrals over the faces of the cube, but the signs + and − placed
before these integrals need to be determined. However, denoting the canonical
coordinates of R
p+1 by t
0 , . . . , t
p gives formulas of type
ω =
p i (t)dt
0
∧ . . .
dt i . . . ∧ dt
p ,
dω =
(−1)
i D i p i (t)dt
0
∧ . . . ∧ dt
p ,
so that the integral of dω is the sum of the integrals over K of the functions
(−1)
i D i p i (t). To calculate them, first integrate with respect to the corresponding t
i s. This gives (FT)
(−1)
i
I p p 1
t
0 , . . . , 1, . . . , t
p
dt
0 . . . dt
i . . . dt
p +
+(−1)
i+1
I p p 1
t
0 , . . . , 0, . . . , t
p
dt
0 . . . dt
i . . . dt
p
involving the extended integrals of ω over the faces of the cube I
p+1 . More
precisely, write F
+
i for the face t i = 1 of the cube and F
−
i for the face t i = 0,
and define the extended integrals of ω over these faces by using the parametric
representation
ϕ
+
i :
t 0 , . . . ,
t i , . . . , t p
−→ (t 0 , . . . , 1, . . . , t p )
(10.31)
in the first case and the the analogous formula in the second. If F is a face
of the cube, set
ε(F ) =
(−1)
i
if F = F
+
i ,
(−1)
i+1 if F = F
−
i .
(10.32)