§ 2. Differential Forms of Degree 1
175
Indeed, start with some x ∈ U . The right hand side can be obtained at this
point from ω(z; dz) by replacing z by g ◦ f (x) = g[f (x)] and dz by
d(g ◦ f )(x; dx) = dg [f (x); f
(x)dx]
in it. Therefore, the right hand side can be calculated by the following
operations: first replace z by g(y) and dz by dg(y; dy), which replaces ω
by ω ◦ g(y; dy), then replace y by f (x) and dy by f
(x)dx, which replaces
ω ◦ g(y; dy) by (ω ◦ g) ◦ f (x; dx), giving (9).
In particular, we may suppose that
27 U = I, so that f is a path μ in V
and g ◦ μ a path in W . Integrating, (8) and (9) immediately show that
g◦μ
ω =
μ
ω ◦ g .
(6.10)
If V is also an interval in R, we recover the fact that, within reasonable
conditions, the value of a curvilinear integral is independent of the chosen
parametrization.
7 – Effect of a Homotopy on an Integral
(i) Differentiation with respect to a path. Like in the very particular case of
holomorphic functions discussed in the previous chapter, the fundamental
property of the integral of a closed (but not necessarily exact) differential
form consists in not changing when deforming the path of integration without changing its endpoints, i.e. by a fixed-endpoint homotopy. Here too, the
arguments of Chapter VIII provide the method and the results.
Before extending what has been proved in Chapter VIII to differential
forms, the differentiation formula with respect to the integration path of
Chapter VIII, n
◦ 3, (ii) must first be generalized. In other words, consider an
admissible path μ : I = [0, 1] −→ G in the open subset G of the Cartesian
space E, and an admissible path ν : I −→ E in E and differentiate the
expression
F (μ + sν) =
μ+sν
ω =
ω [μ(t) + sν(t); μ
(t) + sν
(t)] dt
(7.1)
with respect to s, where s is a parameter varying in a sufficiently small interval
around 0. But as shorthand for (1), write
F (μ + sν) =
ω (μ + sν; μ
) dt + s
ω (μ + sν; ν
) dt
(7.1’)
=
ω (μ + sν; dμ) + s
ω (μ + sν; dν)
or in coordinates,
27 The fact that I is not open is not important.
175
Indeed, start with some x ∈ U . The right hand side can be obtained at this
point from ω(z; dz) by replacing z by g ◦ f (x) = g[f (x)] and dz by
d(g ◦ f )(x; dx) = dg [f (x); f
(x)dx]
in it. Therefore, the right hand side can be calculated by the following
operations: first replace z by g(y) and dz by dg(y; dy), which replaces ω
by ω ◦ g(y; dy), then replace y by f (x) and dy by f
(x)dx, which replaces
ω ◦ g(y; dy) by (ω ◦ g) ◦ f (x; dx), giving (9).
In particular, we may suppose that
27 U = I, so that f is a path μ in V
and g ◦ μ a path in W . Integrating, (8) and (9) immediately show that
g◦μ
ω =
μ
ω ◦ g .
(6.10)
If V is also an interval in R, we recover the fact that, within reasonable
conditions, the value of a curvilinear integral is independent of the chosen
parametrization.
7 – Effect of a Homotopy on an Integral
(i) Differentiation with respect to a path. Like in the very particular case of
holomorphic functions discussed in the previous chapter, the fundamental
property of the integral of a closed (but not necessarily exact) differential
form consists in not changing when deforming the path of integration without changing its endpoints, i.e. by a fixed-endpoint homotopy. Here too, the
arguments of Chapter VIII provide the method and the results.
Before extending what has been proved in Chapter VIII to differential
forms, the differentiation formula with respect to the integration path of
Chapter VIII, n
◦ 3, (ii) must first be generalized. In other words, consider an
admissible path μ : I = [0, 1] −→ G in the open subset G of the Cartesian
space E, and an admissible path ν : I −→ E in E and differentiate the
expression
F (μ + sν) =
μ+sν
ω =
ω [μ(t) + sν(t); μ
(t) + sν
(t)] dt
(7.1)
with respect to s, where s is a parameter varying in a sufficiently small interval
around 0. But as shorthand for (1), write
F (μ + sν) =
ω (μ + sν; μ
) dt + s
ω (μ + sν; ν
) dt
(7.1’)
=
ω (μ + sν; dμ) + s
ω (μ + sν; dν)
or in coordinates,
27 The fact that I is not open is not important.