168
IX – Multivariate Differential and Integral Calculus
p α (η) = ρ
i
α (η)p i (ξ) ,
(4.9)
where the derivative ρ
i
α (η) = dξ
i /dη
α is calculated at the point η(x). We only
mostly argue in terms of standard Cartesian coordinate, but using arbitrary
local chart cannot be avoided when extending the theory to differentiable
manifolds.
Finally note the similarity – and not the identity – between vector fields
and differential forms: a vector field on an open set U ⊂ E associates to
each x ∈ U a vector of E (i.e. is a map from U to E), whereas a differential
form associates to each x ∈ U a covector of E (i.e. is a map from U to the
possibly complex dual E
∗ of E). This is the difference between tensor fields
of type (1, 0) and (0, 1).
5 – Local Primitives
(i) Existence : calculations in terms of coordinates. Having said that, and
considering a Cartesian space E equipped with a basis (a i ), let us return to
the search for a primitive F of a differential form ω = p i (x)dx
i of class C
1 .
We will consider an open connected subset G, i.e. a domain, as the general
case can be reduced to it in an obvious way. In all that follows, x
i will denoted
the coordinates of a point or a vector with respect to the basis taken and we
set D i = d/dx
i as usual.
Relations
D i F = p i for 1 ≤ i ≤ n
(5.1)
obviously require
D j p i − D i p j = 0
(5.2)
for all i and j, which in reality gives n(n − 1)/2 independent relations, for
example those for which i > j ; if these necessary conditions hold, ω is said
to be closed.
Exercise. Show that this definition is independent of the choice of the
basis (a i ). Does relation (2) hold in an arbitrary local chart ?
In the case of holomorphic functions, we know there are always local primitives; this remains true in the general case, a result already known to Euler
for two variable forms. It can be proved as in Chapter VIII, n
◦ 2, (i), formula (2), though not quite so easily.
First, let F be a C
2 function defined at least on the ball B : x < R
centered at O in E [to argue in the neighbourhood of an arbitrary point a,
consider F (a + x)]. For given x ∈ B, the function t → F (tx) is defined on
an open subset of R containing I = [0, 1]. Setting D = d/dt, the multivariate
chain rule shows that
D {F (tx)} = D i F (tx).x
i .
(5.3)
IX – Multivariate Differential and Integral Calculus
p α (η) = ρ
i
α (η)p i (ξ) ,
(4.9)
where the derivative ρ
i
α (η) = dξ
i /dη
α is calculated at the point η(x). We only
mostly argue in terms of standard Cartesian coordinate, but using arbitrary
local chart cannot be avoided when extending the theory to differentiable
manifolds.
Finally note the similarity – and not the identity – between vector fields
and differential forms: a vector field on an open set U ⊂ E associates to
each x ∈ U a vector of E (i.e. is a map from U to E), whereas a differential
form associates to each x ∈ U a covector of E (i.e. is a map from U to the
possibly complex dual E
∗ of E). This is the difference between tensor fields
of type (1, 0) and (0, 1).
5 – Local Primitives
(i) Existence : calculations in terms of coordinates. Having said that, and
considering a Cartesian space E equipped with a basis (a i ), let us return to
the search for a primitive F of a differential form ω = p i (x)dx
i of class C
1 .
We will consider an open connected subset G, i.e. a domain, as the general
case can be reduced to it in an obvious way. In all that follows, x
i will denoted
the coordinates of a point or a vector with respect to the basis taken and we
set D i = d/dx
i as usual.
Relations
D i F = p i for 1 ≤ i ≤ n
(5.1)
obviously require
D j p i − D i p j = 0
(5.2)
for all i and j, which in reality gives n(n − 1)/2 independent relations, for
example those for which i > j ; if these necessary conditions hold, ω is said
to be closed.
Exercise. Show that this definition is independent of the choice of the
basis (a i ). Does relation (2) hold in an arbitrary local chart ?
In the case of holomorphic functions, we know there are always local primitives; this remains true in the general case, a result already known to Euler
for two variable forms. It can be proved as in Chapter VIII, n
◦ 2, (i), formula (2), though not quite so easily.
First, let F be a C
2 function defined at least on the ball B : x < R
centered at O in E [to argue in the neighbourhood of an arbitrary point a,
consider F (a + x)]. For given x ∈ B, the function t → F (tx) is defined on
an open subset of R containing I = [0, 1]. Setting D = d/dt, the multivariate
chain rule shows that
D {F (tx)} = D i F (tx).x
i .
(5.3)