§ 2. Differential Forms of Degree 1
181
So finally,
F (μ + ν) − F (μ) = ω [μ(1); ν(1)] − ω [μ(0); ν(0)] + O
|
|
|ν| |
|
2
.
(7.9)
This result is more than sufficient to justify (7).
But (7) is based on formula (4’), i.e. supposes that ω is closed. If it is not
the case, it is necessary to return to the formula (4) in its entirety. Taking
s = 0 in it, it can be deduced that probably
dF (μ; ν) = ω [μ(1); ν(1)] − ω [μ(0); ν(0)] +
I
dω [μ(t); ν(t), μ
(t)] dt .
(7.10)
Like in the case of the simpler formula (7), we leave it to the reader to check
this formula.
181
So finally,
F (μ + ν) − F (μ) = ω [μ(1); ν(1)] − ω [μ(0); ν(0)] + O
|
|
|ν| |
|
2
.
(7.9)
This result is more than sufficient to justify (7).
But (7) is based on formula (4’), i.e. supposes that ω is closed. If it is not
the case, it is necessary to return to the formula (4) in its entirety. Taking
s = 0 in it, it can be deduced that probably
dF (μ; ν) = ω [μ(1); ν(1)] − ω [μ(0); ν(0)] +
I
dω [μ(t); ν(t), μ
(t)] dt .
(7.10)
Like in the case of the simpler formula (7), we leave it to the reader to check
this formula.