Tangent vector

In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R"n". More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in terms of germs. Formally, a tangent vector at the point formula_1 is a linear derivation of the algebra defined by the set of germs at formula_1.
Motivation.
Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor properties.
Calculus.
Let formula_3 be a parametric smooth curve. The tangent vector is given by formula_4, where we have used a prime instead of the usual dot to indicate differentiation with respect to parameter "t". The unit tangent vector is given by
Example.
Given the curve
in formula_7, the unit tangent vector at formula_8 is given by
Contravariance.
If formula_3 is given parametrically in the "n"-dimensional coordinate system "xi" (here we have used superscripts as an index instead of the usual subscript) by formula_11 or
then the tangent vector field formula_13 is given by
Under a change of coordinates
the tangent vector formula_16 in the "ui"-coordinate system is given by
where we have used the Einstein summation convention. Therefore, a tangent vector of a smooth curve will transform as a contravariant tensor of order one under a change of coordinates.
Definition.
Let formula_18 be a differentiable function and let formula_19 be a vector in formula_20. We define the directional derivative in the formula_19 direction at a point formula_22 by
The tangent vector at the point formula_24 may then be defined as
Properties.
Let formula_26 be differentiable functions, let formula_27 be tangent vectors in formula_20 at formula_22, and let formula_30. Then
Tangent vector on manifolds.
Let formula_34 be a differentiable manifold and let formula_35 be the algebra of real-valued differentiable functions on formula_34. Then the tangent vector to formula_34 at a point formula_1 in the manifold is given by the derivation formula_39 which shall be linear — i.e., for any formula_40 and formula_30 we have
Note that the derivation will by definition have the Leibniz property