§ 2. Differential Forms of Degree 1
169
The FT then shows that
F (x) = F (0) +
1
0
D i F (tx).x
i dt .
(5.4)
Hence, if a closed differential ω = p i (x)dx
i with coefficients of class C
1 –
or even C
0 , but the following arguments fall apart in this case – admits a
primitive F on the ball B where it is defined, then for all x ∈ B, up to an
additive constant
F (x) =
1
0
p i (tx)x
i dt =
1
0
ω(tx; x)dt
(5.5)
necessarily. D j F = p j for all j remains to be check that. For this, the right
hand side of (5) needs to be differentiated with respect to x
j . Now, the p i
being C
1 , the function de (t, x) ∈ I × G being integrated has continuous first
order partial derivatives with respect to the x
i , and so
D j F (x) =
1
0
D j
p i (tx)x
i
dt =
(5.6)
=
1
0
D j {p i (tx)} x
i dt +
1
0
p i (tx)D j x
i dt =
=
1
0
D j p i (tx).tx
i dt +
1
0
p i (tx)δ
i
j dt =
=
1
0
D j p i (tx).tx
i dt +
1
0
p j (tx)dt ,
where δ
i
j = D j x
i . The Kronecker delta already mentioned in (1.8), is equal
to 1 or 0 according to whether i and j are equal or not. But, by (2) and by
(2.20) applied to p j
D j p i (tx).tx
i = D i p j (tx).tx
i = tD {p j (tx)} .
(5.7)
As D = d/dt, integration by parts is possible, and so
D j F (x) = tp j (tx)
1
0
−
1
0
p j (tx)dt +
1
0
p j (tx)dt = p j (x) ,
which solves the problem.
This calculation applies to any star-shaped open set G, i.e. in which there
is some a ∈ G such that, for all x ∈ G, the line segment [a, x] is contained in
G. In a star-shaped open subset
23 of a Cartesian space, all closed differential
forms of class C
1 have a primitive, i.e. are exact. In particular, in a star
domain G ⊂ C, any real harmonic function is the real part of a holomorphic
function on G – a result already shown in Chapter VIII –, namely
23 or simply connected one, as will be shown later.
169
The FT then shows that
F (x) = F (0) +
1
0
D i F (tx).x
i dt .
(5.4)
Hence, if a closed differential ω = p i (x)dx
i with coefficients of class C
1 –
or even C
0 , but the following arguments fall apart in this case – admits a
primitive F on the ball B where it is defined, then for all x ∈ B, up to an
additive constant
F (x) =
1
0
p i (tx)x
i dt =
1
0
ω(tx; x)dt
(5.5)
necessarily. D j F = p j for all j remains to be check that. For this, the right
hand side of (5) needs to be differentiated with respect to x
j . Now, the p i
being C
1 , the function de (t, x) ∈ I × G being integrated has continuous first
order partial derivatives with respect to the x
i , and so
D j F (x) =
1
0
D j
p i (tx)x
i
dt =
(5.6)
=
1
0
D j {p i (tx)} x
i dt +
1
0
p i (tx)D j x
i dt =
=
1
0
D j p i (tx).tx
i dt +
1
0
p i (tx)δ
i
j dt =
=
1
0
D j p i (tx).tx
i dt +
1
0
p j (tx)dt ,
where δ
i
j = D j x
i . The Kronecker delta already mentioned in (1.8), is equal
to 1 or 0 according to whether i and j are equal or not. But, by (2) and by
(2.20) applied to p j
D j p i (tx).tx
i = D i p j (tx).tx
i = tD {p j (tx)} .
(5.7)
As D = d/dt, integration by parts is possible, and so
D j F (x) = tp j (tx)
1
0
−
1
0
p j (tx)dt +
1
0
p j (tx)dt = p j (x) ,
which solves the problem.
This calculation applies to any star-shaped open set G, i.e. in which there
is some a ∈ G such that, for all x ∈ G, the line segment [a, x] is contained in
G. In a star-shaped open subset
23 of a Cartesian space, all closed differential
forms of class C
1 have a primitive, i.e. are exact. In particular, in a star
domain G ⊂ C, any real harmonic function is the real part of a holomorphic
function on G – a result already shown in Chapter VIII –, namely
23 or simply connected one, as will be shown later.