196
IX – Multivariate Differential and Integral Calculus
But the proof of (15) also shows that
D 2 [ω(σ; D 1 σ)] = ω
(σ; D 2 σ, D 1 σ) + ω (σ; D 2 D 1 σ) =
= ω
(σ; D 2 σ, D 1 σ) + ω (σ; D 1 D 2 σ) ,
and so, by (15) and the definition of dω,
D 1 [ω (σ; D 2 σ)] = D 2 [ω (σ; D 1 σ)] + dω (σ; D 1 σ; D 2 σ) .
(9.16)
Hence, by (15), it follows that
F
(s) =
D 2 [ω (σ; D 1 σ)] dt +
dω (σ; D 1 σ, D 2 σ) dt .
(9.17)
As D 2 = d/dt, the first integral is the change of ω(σ; D 1 σ) between t = 0
and t = 1. Integrating F
(s) over (0, 1), we then get (FT)
F (μ 1 ) − F (μ 0 ) =
ω [σ (s, 1) ; D 1 σ (s, 1)] ds −
(9.18)
−
ω [σ (s, 0) ; D 1 σ (s, 0)] ds +
+
I 2
dω (σ; D 1 σ; D 2 σ) .dsdt .
Using the paths μ s and ν t defined above, (18) can be written as
F (μ 1 ) − F (μ 0 ) = F (ν 1 ) − F (ν 0 ) +
dω(. . .) .
(9.19)
If the four integration paths occurring in (19) are concatenated into a single
path ∂σ : [0, 4] −→ G given by the formulas
∂σ(t) =
σ(0, t)
= μ 0 (t)
( 0≤ t ≤ 1)
σ(t − 1, 1) = ν 1 (t − 1) (1 ≤ t ≤ 2)
σ(1, 3 − t) = μ 1 (3 − t) (2 ≤ t ≤ 3)
σ(4 − t, 0) = ν 0 (4 − t) (3 ≤ t ≤ 4) ,
relation (19) becomes
∂σ
ω =
σ
dω
(9.20)
provided, generally speaking, that we set
σ
=
I 2
[σ(s, t); D 1 σ(s, t), D 2 σ(s, t)] dsdt
(9.21)
for any sufficiently differentiable 2-dimensional path σ : I
2
−→ G and any
differential form of degree 2 on G; the similarity with the definition of a