§ 3. Integration of Differential Forms
201
method consisted in using a “ bi-univocal parametric representation ”, i.e.
bijective, of S, given by functions
x = ϕ(s, t) , y = ψ(s, t) , z = θ(s, t)
(9.26)
of two real “ parameters ” s, t varying in a subset A of R
2 bounded by one or
many simple curves such as those encountered in Cauchy’s residue formula;
this amounts to setting
σ(s, t) = (ϕ(s, t), ψ(s, t), θ(s, t)) .
Having said this, dydz, dzdx, dxdy had to be replaced by
D(ψ, θ)
D(s, t)
dsdt ,
D(θ, ϕ)
D(s, t)
dsdt ,
D(ϕ, ψ)
D(s, t)
dsdt
(9.27)
and x, y, z by their expressions in terms of s and t in the function P, Q, R ; an
expression of the form r(s, t)dsdt was thus obtained and integrated over the
domain of variation A of (s, t). To make sure that S was indeed a “ smooth ”
surface, without sharp corners, edges or any other singularities, it was assumed that the above three Jacobians were never simultaneously zero, in
other words, that the tangent linear map to σ at (s, t) was everywhere injective (the necessity of this assumption regarding submanifolds of a Cartesian
space will be explained in n
◦ 13, (iv)) or else that σ was of rank 2; the image
of R
2 under the tangent map σ
(s, t) was then the “ tangent plane ” the the
surface S at the point σ(s, t).
Clearly, the expression P dydz + Qdzdx + Rdxdy to be integrated is just
the differential form
ω = P dy ∧ dz + Qdz ∧ dx + Rdx ∧ dy
on R
3 and, if K is the unit square I
2 , then the surface integral to be computed
is just the integral (21’) of ω extended to the path σ.
The integral thus defined also had to be shown to depend solely on the
given vector field and surface S, and not on the chosen parametric representation; as will be seen in the next n
◦ , this was ensured by a change of variable
formula in ordinary double integrals, the only difficulty residing in its proof.
As for (20), it is the traditional Stokes formula; it was applied by assuming that the vector field (P, Q, R) to be integrated was the rotational of a
vector field H. The integral of H along the curve C bounding S gave the
left hand side of (20). The expression to be integrated was often described
as the scalar product of H and of an infinitesimal vector with components
dx, dy, dz, written dM , where the letter M denotes a variable point of the
curve. Physicists wrote the double integral in a similar way by regarding expressions (27) as the components of the metaphysical vector dS normal to S
(i.e. orthogonal to the tangent plane to S) at the point considered and with
length “ the element of infinitesimal surface ” of S, whatever that meant, so
that (20) finally became
201
method consisted in using a “ bi-univocal parametric representation ”, i.e.
bijective, of S, given by functions
x = ϕ(s, t) , y = ψ(s, t) , z = θ(s, t)
(9.26)
of two real “ parameters ” s, t varying in a subset A of R
2 bounded by one or
many simple curves such as those encountered in Cauchy’s residue formula;
this amounts to setting
σ(s, t) = (ϕ(s, t), ψ(s, t), θ(s, t)) .
Having said this, dydz, dzdx, dxdy had to be replaced by
D(ψ, θ)
D(s, t)
dsdt ,
D(θ, ϕ)
D(s, t)
dsdt ,
D(ϕ, ψ)
D(s, t)
dsdt
(9.27)
and x, y, z by their expressions in terms of s and t in the function P, Q, R ; an
expression of the form r(s, t)dsdt was thus obtained and integrated over the
domain of variation A of (s, t). To make sure that S was indeed a “ smooth ”
surface, without sharp corners, edges or any other singularities, it was assumed that the above three Jacobians were never simultaneously zero, in
other words, that the tangent linear map to σ at (s, t) was everywhere injective (the necessity of this assumption regarding submanifolds of a Cartesian
space will be explained in n
◦ 13, (iv)) or else that σ was of rank 2; the image
of R
2 under the tangent map σ
(s, t) was then the “ tangent plane ” the the
surface S at the point σ(s, t).
Clearly, the expression P dydz + Qdzdx + Rdxdy to be integrated is just
the differential form
ω = P dy ∧ dz + Qdz ∧ dx + Rdx ∧ dy
on R
3 and, if K is the unit square I
2 , then the surface integral to be computed
is just the integral (21’) of ω extended to the path σ.
The integral thus defined also had to be shown to depend solely on the
given vector field and surface S, and not on the chosen parametric representation; as will be seen in the next n
◦ , this was ensured by a change of variable
formula in ordinary double integrals, the only difficulty residing in its proof.
As for (20), it is the traditional Stokes formula; it was applied by assuming that the vector field (P, Q, R) to be integrated was the rotational of a
vector field H. The integral of H along the curve C bounding S gave the
left hand side of (20). The expression to be integrated was often described
as the scalar product of H and of an infinitesimal vector with components
dx, dy, dz, written dM , where the letter M denotes a variable point of the
curve. Physicists wrote the double integral in a similar way by regarding expressions (27) as the components of the metaphysical vector dS normal to S
(i.e. orthogonal to the tangent plane to S) at the point considered and with
length “ the element of infinitesimal surface ” of S, whatever that meant, so
that (20) finally became