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IX – Multivariate Differential and Integral Calculus
14 – Vector Fields and Differential Operators
Since we have defined tensors at a point of a manifold X of class C
r in n
◦ 12,
(i), we can now define tensor fields of type (p, q) on X or, more generally (?),
on an open subset of X. They associate to each point x ∈ X a tensor of type
(p, q) at the point x considered, either as a multilinear function of p vectors
and q covectors at x, or as a system of numbers T
kh...
ij... depending on x and
on a local chart at x, and compelled to transform by the Italian formulas
under chart change. T will be said to be of class C
s if in every local chart, its
components are C
s functions on the corresponding open Cartesian subspace;
we need to assume that s ≤ r − 1 because if changes of charts are of class C
r ,
their first partial derivatives are only of class C
r−1 ; thus there is no hope of
the Italian formulas transforming C
r functions into C
r functions.
In practice, the most important tensor fields are the vector fields and
the differential forms that will be defined below. A vector field L is of type
(0, 1), so that it is obtained by associating to each x ∈ X a tangent vector
L(x) ∈ X
(x) to X at x. Hence, if (U, ϕ) is a chart, then for each x ∈ U there
is a vector
L(x)(ϕ) = L
i (ξ)e i whereξ = ϕ(x) ,
i.e. a vector field (in the sense used by physicists) on the open subset ϕ(U )
of R
d , d = dim(X) with the obvious formulas for chart change. L will also be
said to be of class C
s if so are the functions L
i .
If f is function of class C
s (1 ≤ s ≤ r) on an open set W ⊂ X, then set
Lf (x) = df [x; L(x)]
for all x ∈ X ; many authors prefer to denote the function Lf by D L f . If
(U, ϕ) is a chart, with U ⊂ W , and if F is the function which expresses f in
ϕ(U ), so that f = F ◦ ϕ, then, by the multivariate chain rule,
Lf (x) = dF [ϕ(x); ϕ
(x)L(x)] ;
but by (12.13), we know that for h ∈ X
(x),
ϕ
(x)h = h(ϕ) = h
i (ϕ)e i .
Setting ξ = ϕ(x), it follows that
Lf (x) = dF
ξ; L
i (ξ)e i
= L
i (ξ)dF (ξ; e i ) = L
i (ξ)D i F (ξ) ,
(14.1)
where the D i F are the usual partial derivatives of F and the L
i (x) the components of L(x) in the chart considered. Obviously,
L(fg) = Lf.g + f.Lg
for all f and g.
IX – Multivariate Differential and Integral Calculus
14 – Vector Fields and Differential Operators
Since we have defined tensors at a point of a manifold X of class C
r in n
◦ 12,
(i), we can now define tensor fields of type (p, q) on X or, more generally (?),
on an open subset of X. They associate to each point x ∈ X a tensor of type
(p, q) at the point x considered, either as a multilinear function of p vectors
and q covectors at x, or as a system of numbers T
kh...
ij... depending on x and
on a local chart at x, and compelled to transform by the Italian formulas
under chart change. T will be said to be of class C
s if in every local chart, its
components are C
s functions on the corresponding open Cartesian subspace;
we need to assume that s ≤ r − 1 because if changes of charts are of class C
r ,
their first partial derivatives are only of class C
r−1 ; thus there is no hope of
the Italian formulas transforming C
r functions into C
r functions.
In practice, the most important tensor fields are the vector fields and
the differential forms that will be defined below. A vector field L is of type
(0, 1), so that it is obtained by associating to each x ∈ X a tangent vector
L(x) ∈ X
(x) to X at x. Hence, if (U, ϕ) is a chart, then for each x ∈ U there
is a vector
L(x)(ϕ) = L
i (ξ)e i whereξ = ϕ(x) ,
i.e. a vector field (in the sense used by physicists) on the open subset ϕ(U )
of R
d , d = dim(X) with the obvious formulas for chart change. L will also be
said to be of class C
s if so are the functions L
i .
If f is function of class C
s (1 ≤ s ≤ r) on an open set W ⊂ X, then set
Lf (x) = df [x; L(x)]
for all x ∈ X ; many authors prefer to denote the function Lf by D L f . If
(U, ϕ) is a chart, with U ⊂ W , and if F is the function which expresses f in
ϕ(U ), so that f = F ◦ ϕ, then, by the multivariate chain rule,
Lf (x) = dF [ϕ(x); ϕ
(x)L(x)] ;
but by (12.13), we know that for h ∈ X
(x),
ϕ
(x)h = h(ϕ) = h
i (ϕ)e i .
Setting ξ = ϕ(x), it follows that
Lf (x) = dF
ξ; L
i (ξ)e i
= L
i (ξ)dF (ξ; e i ) = L
i (ξ)D i F (ξ) ,
(14.1)
where the D i F are the usual partial derivatives of F and the L
i (x) the components of L(x) in the chart considered. Obviously,
L(fg) = Lf.g + f.Lg
for all f and g.