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 Rn.
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
is a linear derivation of the algebra defined by the set of germs at
Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor properties.
be a parametric smooth curve.
The tangent vector is given by
provided it exists and provided
, where we have used a prime instead of the usual dot to indicate differentiation with respect to parameter t.[1] The unit tangent vector is given by
, the unit tangent vector at
is given parametrically in the n-dimensional coordinate system xi (here we have used superscripts as an index instead of the usual subscript) by
then the tangent vector field
Under a change of coordinates
the tangent vector
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.
be a differentiable function and let
We define the directional derivative in the
direction at a point
The tangent vector at the point
be differentiable functions, let
be tangent vectors in
be a differentiable manifold and let
be the algebra of real-valued differentiable functions on
Then the tangent vector to
in the manifold is given by the derivation
we have Note that the derivation will by definition have the Leibniz property