§ 3. Integration of Differential Forms
183
But as already mentioned in n
◦ 1, (ii), this identification has no intrinsic or
absolute meaning, whether in a general vector space lacking a distinguished
scalar product, or in an Euclidean space, when non-rectangular coordinate
systems are used.
Moreover, defining the “ gradient ” of a function F as we have done, i.e.
by differentiating F (x + th) at t = 0, is perfectly natural even and especially
from a physics viewpoint. For example what is a “ temperature gradient ”
but a measure of the rate of change of temperature when going from a point
x to a point x + th in the direction defined by a given vector h? There
are no coordinates in this definition, and a “ rate of change ” has always
been a derivative. In this case, the physical reason for writing dF (x; h) as
(grad F (x)|h) is to highlight the direction, that of the vector grad F (x), in
which the temperature change in the neighbourhood of x is fastest.
Remaining in dimension 3, there are three independent conditions (5.14’);
as 3 is the unique number for which n(n − 1)/2 = n, the Creator must have
invented this miracle on the eve of the Big Bang to mystify his creatures into
believing that they would thus more easily understand his Complete Works
than if he had, for example, chosen a four-dimensional space. This leads
physicists to wrongly associate to the vector field (P, Q, R), the vector field
(D 2 R − D 3 Q, D 3 P − D 1 R, D 1 Q − D 2 P ), that they again call the rotational
of the given vector field; its cancelation is a necessary (and sufficient in a
simply connected domain) condition for the vector field (P, Q, R) to come
from a potential (Theorem 3). But for n = 4, the Relativity space that the
mystified eventually discovered, there are already 6 conditions (5.14’) and it
is no longer a question of rotationals in the sense of a vector field. For n > 4
this is even less so.
(ii) Differential forms of degree 2. The proper generalization of physicists’
rotational is the exterior derivative of a differential form, which have already
cropped up in n
◦ 5 at the end of the previous n
◦ :
dω(x; h, k) = ω
(x; h, k) − ω
(x; k, h) .
In dimension 3, if ω = P dx + Qdy + Rdz, writing (5.15) explicitly, the expression dω(x; h, k) is indeed equal to
(D 2 R − D 3 Q)
h
2 k
3
− h
3 k
2
+ (D 3 P − D 1 R)
h
3 k
1
− h
1 k
3
+ (D 1 Q − D 2 P )
h
1 k
2
− h
2 k
1
,
which gives rise to the components of the physicists’ rotational.
As a function of h and k for a given x, dω(x; h, k) is an alternating bilinear
form in h, k. Its generalization is a function ω(x; h, k) of x ∈ G and of two
in Chapter II rational numbers were identified to real ones) when the objects
identified satisfy for example the same computation rules. But this is highly
questionable when for example it suppose a particular choice of a coordinate
system.
183
But as already mentioned in n
◦ 1, (ii), this identification has no intrinsic or
absolute meaning, whether in a general vector space lacking a distinguished
scalar product, or in an Euclidean space, when non-rectangular coordinate
systems are used.
Moreover, defining the “ gradient ” of a function F as we have done, i.e.
by differentiating F (x + th) at t = 0, is perfectly natural even and especially
from a physics viewpoint. For example what is a “ temperature gradient ”
but a measure of the rate of change of temperature when going from a point
x to a point x + th in the direction defined by a given vector h? There
are no coordinates in this definition, and a “ rate of change ” has always
been a derivative. In this case, the physical reason for writing dF (x; h) as
(grad F (x)|h) is to highlight the direction, that of the vector grad F (x), in
which the temperature change in the neighbourhood of x is fastest.
Remaining in dimension 3, there are three independent conditions (5.14’);
as 3 is the unique number for which n(n − 1)/2 = n, the Creator must have
invented this miracle on the eve of the Big Bang to mystify his creatures into
believing that they would thus more easily understand his Complete Works
than if he had, for example, chosen a four-dimensional space. This leads
physicists to wrongly associate to the vector field (P, Q, R), the vector field
(D 2 R − D 3 Q, D 3 P − D 1 R, D 1 Q − D 2 P ), that they again call the rotational
of the given vector field; its cancelation is a necessary (and sufficient in a
simply connected domain) condition for the vector field (P, Q, R) to come
from a potential (Theorem 3). But for n = 4, the Relativity space that the
mystified eventually discovered, there are already 6 conditions (5.14’) and it
is no longer a question of rotationals in the sense of a vector field. For n > 4
this is even less so.
(ii) Differential forms of degree 2. The proper generalization of physicists’
rotational is the exterior derivative of a differential form, which have already
cropped up in n
◦ 5 at the end of the previous n
◦ :
dω(x; h, k) = ω
(x; h, k) − ω
(x; k, h) .
In dimension 3, if ω = P dx + Qdy + Rdz, writing (5.15) explicitly, the expression dω(x; h, k) is indeed equal to
(D 2 R − D 3 Q)
h
2 k
3
− h
3 k
2
+ (D 3 P − D 1 R)
h
3 k
1
− h
1 k
3
+ (D 1 Q − D 2 P )
h
1 k
2
− h
2 k
1
,
which gives rise to the components of the physicists’ rotational.
As a function of h and k for a given x, dω(x; h, k) is an alternating bilinear
form in h, k. Its generalization is a function ω(x; h, k) of x ∈ G and of two
in Chapter II rational numbers were identified to real ones) when the objects
identified satisfy for example the same computation rules. But this is highly
questionable when for example it suppose a particular choice of a coordinate
system.