§ 2. Differential Forms of Degree 1
167
ω = p 1 (x)dx
1 . . . + p n (x)dx
n = p i dx
i ,
(4.5)
where the p i are given possibly complex or even vector-valued functions of
class C
1 at least on U , and where the x
i are the coordinates of x ∈ U with
respect to the given basis of E; this definition has no “ absolute ” meaning if
the changes undergone by the p i under a change of basis is not made clear,
but a direct definition of differential forms can be given, which gets rid of
this problem.
Indeed, if there a function F of class at least C
1 on U , its differential is
a function dF (x; h) of a point x ∈ U and of a vector h ∈ E. For fixed x,
this function is linear in h. The natural generalization is therefore to consider
functions ω(x; h) subject to the same conditions, in other words tensor fields
of type (0, 1) or – what amounts to the same since a linear functional
ω(x) : h −→ ω(x; h)
on E, i.e. an element of E
∗ is associated to each x ∈ U – maps from U to E
∗
(or to E
∗
C since complex valued forms are also considered). Conversely, once
a basis for E is chosen, thanks to this definition, any map ω : U −→ E
∗ can
be written in the form (5). To see this, first write that
ω(x; h) = ω
x; h
i a i
= h
i ω (x; a i ) = p i (x)h
i where p i (x) = ω(x; a i ) .
(4.6)
The expression p i (x)dx
i is then justified by precisely the same reasons as
those given at the end of section (i) of n
◦ 2 : write that h
i = dx
i (x; h).
Conversely, it suffices to associate the function
ω(x; h) = p i (x)h
i
(x ∈ U , h ∈ E)
(4.7)
to (5) to reduce to the new definition.
As in n
◦ 3, (ii), ω could be written in an arbitrary local chart (U, ϕ). The
latter defines a basis (a i (ξ)) of E at each x ∈ U , where ξ = ϕ(x) ; as seen in
(3.4), any vector h ∈ E can then be written
h = h
i (ξ)a i (ξ) = a i (ξ)dξ
i (x; h)
with coordinates depending both on the chart and the point x considered.
Hence
ω(x; h) = p i (ξ)h
i (ξ) = p i (ξ)dξ
i (x; h) ,
where the p i (x) = ω[x; a i (ξ)] are the components of the tensor field ω in
the chart considered; so, following Leibniz, the expression for ω in the chart
(U, ϕ) is
ω(x; dx) = p i dξ
i .
(4.8)
Under chart change, the coefficients p i (ξ) are transformed like those of a
tensor of type (0, 1) :
167
ω = p 1 (x)dx
1 . . . + p n (x)dx
n = p i dx
i ,
(4.5)
where the p i are given possibly complex or even vector-valued functions of
class C
1 at least on U , and where the x
i are the coordinates of x ∈ U with
respect to the given basis of E; this definition has no “ absolute ” meaning if
the changes undergone by the p i under a change of basis is not made clear,
but a direct definition of differential forms can be given, which gets rid of
this problem.
Indeed, if there a function F of class at least C
1 on U , its differential is
a function dF (x; h) of a point x ∈ U and of a vector h ∈ E. For fixed x,
this function is linear in h. The natural generalization is therefore to consider
functions ω(x; h) subject to the same conditions, in other words tensor fields
of type (0, 1) or – what amounts to the same since a linear functional
ω(x) : h −→ ω(x; h)
on E, i.e. an element of E
∗ is associated to each x ∈ U – maps from U to E
∗
(or to E
∗
C since complex valued forms are also considered). Conversely, once
a basis for E is chosen, thanks to this definition, any map ω : U −→ E
∗ can
be written in the form (5). To see this, first write that
ω(x; h) = ω
x; h
i a i
= h
i ω (x; a i ) = p i (x)h
i where p i (x) = ω(x; a i ) .
(4.6)
The expression p i (x)dx
i is then justified by precisely the same reasons as
those given at the end of section (i) of n
◦ 2 : write that h
i = dx
i (x; h).
Conversely, it suffices to associate the function
ω(x; h) = p i (x)h
i
(x ∈ U , h ∈ E)
(4.7)
to (5) to reduce to the new definition.
As in n
◦ 3, (ii), ω could be written in an arbitrary local chart (U, ϕ). The
latter defines a basis (a i (ξ)) of E at each x ∈ U , where ξ = ϕ(x) ; as seen in
(3.4), any vector h ∈ E can then be written
h = h
i (ξ)a i (ξ) = a i (ξ)dξ
i (x; h)
with coordinates depending both on the chart and the point x considered.
Hence
ω(x; h) = p i (ξ)h
i (ξ) = p i (ξ)dξ
i (x; h) ,
where the p i (x) = ω[x; a i (ξ)] are the components of the tensor field ω in
the chart considered; so, following Leibniz, the expression for ω in the chart
(U, ϕ) is
ω(x; dx) = p i dξ
i .
(4.8)
Under chart change, the coefficients p i (ξ) are transformed like those of a
tensor of type (0, 1) :