§ 2. Differential Forms of Degree 1
177
Substituting this result in (3), it follows that
d
ds
F (μ + sν) =
1
0
{ω
(μ + sν; ν, μ
+ sν
) − ω
(μ + sν; μ
+ sν
, ν)} dt +
+ ω(μ + sν; ν)
t=1
t=0
.
Hence, using the exterior derivative
dω(x; h, k) = ω
(x; h, k) − ω
(x; k, h)
introduced in (5.14), we get
d
ds
F (μ + sν) = ω(μ + sν; ν)
t=1
t=0
+
1
0
dω (μ + sν; ν, μ
+ sν
) dt .
(7.4)
If the form ω is closed, then dω = 0 as already seen in n
◦ 5 and (4) follows
from the relation
d
ds
F (μ + sν) = ω [μ(t) + sν(t); ν(t)]
t=1
t=0
(7.4’)
which generalizes formula (3.5) of Chapter VIII.
(ii) Effect of a homotopy on an integral. The same consequence as in the
previous chapter follow from (4’). If in the open set G where ω is defined,
there are two sufficiently near admissible paths μ 0 and μ 1 for the path
t −→ (1 − s)μ 0 (t) + sμ 1 (t) = μ 0 (t) + s [μ 1 (t) − μ 0 (t)]
to be in G for all s ∈ [0, 1], then formula (4’) applies for μ = μ 0 and ν =
μ 1 − μ 0 . The function F (μ + sν) is, therefore, constant, in both usual cases:
μ 0 and μ 1 have the same endpoints or else are both closed. In the general
case of two homotopic paths, as in Chapter VIII, replace the given homotopy
by a succession of linear homotopies. This gives the result:
Theorem 3. Let G be a domain in a Cartesian space E and ω a closed
differential form of class C
1 on G. The integrals of ω along the two admissible
paths μ 0 and μ 1 in G are equal if one of the following conditions holds :
(a) There is a fixed-endpoint homotopy from μ 0 to μ 1 on G ;
(b) μ 0 and μ 1 are closed homotopic as closed paths in G.
From this, we can deduce that if all closed paths in G are homotopic to
a constant path, then any closed form on G has a primitive; a domain with
this property is said to be simply connected.
Homotopy being an equivalence relation, it is then obvious that condition
(b) of the theorem always holds. The same is true for condition (a); to see
177
Substituting this result in (3), it follows that
d
ds
F (μ + sν) =
1
0
{ω
(μ + sν; ν, μ
+ sν
) − ω
(μ + sν; μ
+ sν
, ν)} dt +
+ ω(μ + sν; ν)
t=1
t=0
.
Hence, using the exterior derivative
dω(x; h, k) = ω
(x; h, k) − ω
(x; k, h)
introduced in (5.14), we get
d
ds
F (μ + sν) = ω(μ + sν; ν)
t=1
t=0
+
1
0
dω (μ + sν; ν, μ
+ sν
) dt .
(7.4)
If the form ω is closed, then dω = 0 as already seen in n
◦ 5 and (4) follows
from the relation
d
ds
F (μ + sν) = ω [μ(t) + sν(t); ν(t)]
t=1
t=0
(7.4’)
which generalizes formula (3.5) of Chapter VIII.
(ii) Effect of a homotopy on an integral. The same consequence as in the
previous chapter follow from (4’). If in the open set G where ω is defined,
there are two sufficiently near admissible paths μ 0 and μ 1 for the path
t −→ (1 − s)μ 0 (t) + sμ 1 (t) = μ 0 (t) + s [μ 1 (t) − μ 0 (t)]
to be in G for all s ∈ [0, 1], then formula (4’) applies for μ = μ 0 and ν =
μ 1 − μ 0 . The function F (μ + sν) is, therefore, constant, in both usual cases:
μ 0 and μ 1 have the same endpoints or else are both closed. In the general
case of two homotopic paths, as in Chapter VIII, replace the given homotopy
by a succession of linear homotopies. This gives the result:
Theorem 3. Let G be a domain in a Cartesian space E and ω a closed
differential form of class C
1 on G. The integrals of ω along the two admissible
paths μ 0 and μ 1 in G are equal if one of the following conditions holds :
(a) There is a fixed-endpoint homotopy from μ 0 to μ 1 on G ;
(b) μ 0 and μ 1 are closed homotopic as closed paths in G.
From this, we can deduce that if all closed paths in G are homotopic to
a constant path, then any closed form on G has a primitive; a domain with
this property is said to be simply connected.
Homotopy being an equivalence relation, it is then obvious that condition
(b) of the theorem always holds. The same is true for condition (a); to see