200
IX – Multivariate Differential and Integral Calculus
These four simple oriented integrals correspond to the obvious four paths
making up ∂σ . As for the double integral of
dω = (D 1 q − D 2 p) dx ∧ dy = r(x, y)dx ∧ dy
occurring in (20), it is calculated by replacing (x, y) by σ(s, t) and dx ∧ dy
by J σ (s, t)dsdt, where
J σ (s, t) =
0 f 1 (t) − f 0 (t)
1
?
= f 0 (t) − f 1 (t)
is the Jacobian of map (24). Here, integral (21’) can, therefore, be written
I 2
r [t, (1 − s)f 0 (t) + sf 1 (t)] . [f 0 (t) − f 1 (t)] .dsdt .
Replacing t by x and making a change of variable y
=
(1 − s)f 0 (x) + sf 1 (x) in the integration with respect to s, we get the integral
1
0
dx
f1(x)
f0(x)
r(x, y)dy ,
where the integral with respect to y is oriented. If the Lebesgue-Fubini formula (33.7) of Chapter V, § 9 is naively applied, the result seems to be just
the extended ordinary double integral of r over the compact subset A of R
2
bounded by the graphs of f 0 and f 1 and the verticals x = 0 and x = 1. This is
the case only if f 0 (t) ≤ f 1 (t) for all t. In fact, denoting by A + (resp. A − ) the
subset of A on which the Jacobian f 1 (x) − f 0 (x) is positive (resp. negative),
(20) can be written as
∂σ
pdx + qdy =
A+
(D 1 q − D 2 p) dxdy −
(9.25)
−
A−
(D 1 q − D 2 p) dxdy .
This corresponds to the fact the the path ∂σ, which is the image under σ of
the border of the square I
2 , is made up of several closed simple curves, some
oriented counterclockwise, others clockwise. If f 1 (t) ≥ f 0 (t) everywhere, we
recover the Green-Riemann formula in a slightly more general case, but as
simple to prove directly with the help of a small Cauchy calculation and of
the most elementary version of Lebesgue-Fubini.
(v) Classical version. In classical analysis, there was a “ surface ” S –
a sphere, a torus, a somewhat deformed rectangle, etc. – and a vector field
(P, Q, R) in the usual three dimensional Euclidean space; the aim was to
define the extended integral
P dydz + Qdzdx + Rdxdy over S. The general
IX – Multivariate Differential and Integral Calculus
These four simple oriented integrals correspond to the obvious four paths
making up ∂σ . As for the double integral of
dω = (D 1 q − D 2 p) dx ∧ dy = r(x, y)dx ∧ dy
occurring in (20), it is calculated by replacing (x, y) by σ(s, t) and dx ∧ dy
by J σ (s, t)dsdt, where
J σ (s, t) =
0 f 1 (t) − f 0 (t)
1
?
= f 0 (t) − f 1 (t)
is the Jacobian of map (24). Here, integral (21’) can, therefore, be written
I 2
r [t, (1 − s)f 0 (t) + sf 1 (t)] . [f 0 (t) − f 1 (t)] .dsdt .
Replacing t by x and making a change of variable y
=
(1 − s)f 0 (x) + sf 1 (x) in the integration with respect to s, we get the integral
1
0
dx
f1(x)
f0(x)
r(x, y)dy ,
where the integral with respect to y is oriented. If the Lebesgue-Fubini formula (33.7) of Chapter V, § 9 is naively applied, the result seems to be just
the extended ordinary double integral of r over the compact subset A of R
2
bounded by the graphs of f 0 and f 1 and the verticals x = 0 and x = 1. This is
the case only if f 0 (t) ≤ f 1 (t) for all t. In fact, denoting by A + (resp. A − ) the
subset of A on which the Jacobian f 1 (x) − f 0 (x) is positive (resp. negative),
(20) can be written as
∂σ
pdx + qdy =
A+
(D 1 q − D 2 p) dxdy −
(9.25)
−
A−
(D 1 q − D 2 p) dxdy .
This corresponds to the fact the the path ∂σ, which is the image under σ of
the border of the square I
2 , is made up of several closed simple curves, some
oriented counterclockwise, others clockwise. If f 1 (t) ≥ f 0 (t) everywhere, we
recover the Green-Riemann formula in a slightly more general case, but as
simple to prove directly with the help of a small Cauchy calculation and of
the most elementary version of Lebesgue-Fubini.
(v) Classical version. In classical analysis, there was a “ surface ” S –
a sphere, a torus, a somewhat deformed rectangle, etc. – and a vector field
(P, Q, R) in the usual three dimensional Euclidean space; the aim was to
define the extended integral
P dydz + Qdzdx + Rdxdy over S. The general