Let
be a
differentiable manifold,
and let
be the algebra of smooth real-valued functions on it. A tangent vector to
at
is an
-linear
mapping
such that
For this definition one can equally well (in fact,
better) use the ring of germs of smooth functions
on
at
.
The tangent vectors to
at
form a vector space over
of dimension
.
It is denoted by
.
Let
,
,
where
is a system of coordinates on
near
.
The
-th
partial derivative at
with respect to
is the tangent vector
where the right hand-side is the usual partial derivative of the function
in the variables
,
at the point
.
One has
(the Kronecker delta) and the
form a basis for
.
This basis for
determined by the coordinate system
is often denoted by
.
A
cotangent vector
at
is an
-linear
mapping
such that the
cotangent space
at
is the dual vector space to
.
The dual basis to
is denoted by
.
One has
The disjoint union
of the tangent spaces
,
,
together with the projection
,
if
,
can be given the structure of a differentiable
vector bundle,
the
tangent bundle.
Similarly, the cotangent spaces
form a vector bundle
dual to
,
called the
cotangent bundle.
The sections of
are the
vector fields
on
,
the sections of
are
differentiable
-forms
on
.
Let
be a mapping of differentiable manifolds and let
be the induced mapping
.
For a tangent vector
at
,
composition with
gives an
-linear
mapping
which is a tangent vector to
at
.
This defines a homomorphism of vector spaces
and a vector bundle morphism
.
In case
and
with global coordinates
and
,
respectively,
is given by
differentiable functions and at each
,
so that the matrix of
with respect to the basis
of
and the basis
of
is given by the
Jacobi matrix
of
at
.
Now, let
be an imbedded manifold. Let
,
be a smooth curve in
,
.
Then
All tangent vectors in
arise in this way. Identifying the vector
(a2)
with the
-vector
,
viewed as a directed line segment starting in
,
one recovers the intuitive picture of the tangent space
as the
-plane
in
tangent to
in
.
A
vector field
on a manifold
can be defined as a derivation (cf.
Derivation in a ring)
in the
-algebra
,
.
Composition with the evaluation mappings
,
,
yields a family of tangent vectors
,
so that
"becomes"
a section of the
tangent bundle.
Given local coordinates
,
can locally be written as
and if a function
in local coordinates is given by
,
then
is the function given in local coordinates by the expression
showing once more the convenience of the
"
/
x"
notation for tangent vectors. (Of course, in practice one
uses a bit more abuse of notation and writes
instead of
.)
Let
be the
-algebra
of germs of smooth functions at
(cf.
Germ).
Let
be the ideal of germs that vanish at zero, and
the ideal generated by all products
for
.
If
are local coordinates at
so that
,
is generated as an ideal in
by
,
and
by the
,
.
In fact, the quotient ring
is the power series ring in
variables over
.
Here
is the ideal of
flat function germs.
(A smooth function is
flat
at a point if it vanishes there with all its derivatives (an example is
at
);
the
"Taylor expansion mapping"
is surjective, a very special consequence of the
Whitney extension theorem.)
Now, let
be a tangent vector of
at
.
Then
by
(a1)
for all constant functions in
.
Also
,
again by
(a1).
Thus, each
defines an element in
,
which is of dimension
because
has dimension
(and that element uniquely determines
).
Moreover, the tangent vectors
clearly define
linearly independent elements in
(because
).
Thus,
the dual space of
.
This point of view is more generally applicable and serves
as the definition of tangent space in analytic and algebraic geometry, cf.
Analytic space;
Zariski tangent space.