172
IX – Multivariate Differential and Integral Calculus
6 – Integration Along a Path. Inverse Images
(i) Integrals of a differential form. Everything that has been said in Chapter
VIII about integrals of holomorphic functions generalizes, with some small
adjustments, to differential forms. So we will remain brief.
Consider a differential form ω = p i (x)dx
i of degree 1 on a domain G
in a finite-dimensional vector space E and assume it has a primitive F on
G. Connect a ∈ G to b ∈ G by a path μ : [0, 1] = I −→ G of class C
1
or, more generally, an admissible path or of class C
1/2 , i.e. such that the
coordinates μ
i (t) of μ(t) are primitives of regulated functions. Writing D
for the differentiation operator with respect to t, the multivariate chain rule
(2.20) shows that outside a countable set of values of t where μ(t) is not
differentiable (i.e. has different right and left derivatives),
D {F [μ(t)]} = dF [μ(t); μ
(t)] = D i F [μ(t)] .Dμ
i (t) = p i [μ(t)] Dμ
i (t) ,
(6.1)
where p i = D i F . The result is a regulated function of t since the p i [μ(t)] are
continuous and the Dμ
i (t) are regulated; hence (FT)
F (b) − F (a) =
I
p i [μ(t)] Dμ
i (t)dt =
I
ω [μ(t); μ
(t)] dt .
(6.2)
The right hand side of (2), which is well-defined for any form ω, is by definition
the integral of ω along μ, denoted by any one of the following
μ
ω =
μ
p i (x)dx
i =
I
ω [μ(t), μ
(t)] dt =
I
p i [μ(t)] dμ
i (t)
(6.3)
with, finally, Stieltjes integrals as in Chapter VIII, n
◦ 2, (iii).
As before, these calculations show that a closed form of class C
1 has a
primitive on G if and only if its integral along a path does not depend on its
endpoints or, equivalently, that its integral along any closed path is zero. In
fact, the result can be generalized to all forms of class C
0 in the following
manner.
Assume that the stated condition holds; then for any a, b ∈ G, we can
write
b
a
ω
for the integral of ω along any path connecting a to b in G; there is no
ambiguity. The traditional formula (Chap. V, § 3, n
◦ 12)
b
a
=
c
a
+
b
c
continues to hold since it is possible to go from a to b through c. . . Having
said this, set
IX – Multivariate Differential and Integral Calculus
6 – Integration Along a Path. Inverse Images
(i) Integrals of a differential form. Everything that has been said in Chapter
VIII about integrals of holomorphic functions generalizes, with some small
adjustments, to differential forms. So we will remain brief.
Consider a differential form ω = p i (x)dx
i of degree 1 on a domain G
in a finite-dimensional vector space E and assume it has a primitive F on
G. Connect a ∈ G to b ∈ G by a path μ : [0, 1] = I −→ G of class C
1
or, more generally, an admissible path or of class C
1/2 , i.e. such that the
coordinates μ
i (t) of μ(t) are primitives of regulated functions. Writing D
for the differentiation operator with respect to t, the multivariate chain rule
(2.20) shows that outside a countable set of values of t where μ(t) is not
differentiable (i.e. has different right and left derivatives),
D {F [μ(t)]} = dF [μ(t); μ
(t)] = D i F [μ(t)] .Dμ
i (t) = p i [μ(t)] Dμ
i (t) ,
(6.1)
where p i = D i F . The result is a regulated function of t since the p i [μ(t)] are
continuous and the Dμ
i (t) are regulated; hence (FT)
F (b) − F (a) =
I
p i [μ(t)] Dμ
i (t)dt =
I
ω [μ(t); μ
(t)] dt .
(6.2)
The right hand side of (2), which is well-defined for any form ω, is by definition
the integral of ω along μ, denoted by any one of the following
μ
ω =
μ
p i (x)dx
i =
I
ω [μ(t), μ
(t)] dt =
I
p i [μ(t)] dμ
i (t)
(6.3)
with, finally, Stieltjes integrals as in Chapter VIII, n
◦ 2, (iii).
As before, these calculations show that a closed form of class C
1 has a
primitive on G if and only if its integral along a path does not depend on its
endpoints or, equivalently, that its integral along any closed path is zero. In
fact, the result can be generalized to all forms of class C
0 in the following
manner.
Assume that the stated condition holds; then for any a, b ∈ G, we can
write
b
a
ω
for the integral of ω along any path connecting a to b in G; there is no
ambiguity. The traditional formula (Chap. V, § 3, n
◦ 12)
b
a
=
c
a
+
b
c
continues to hold since it is possible to go from a to b through c. . . Having
said this, set