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.
Let be a parametric smooth curve. The tangent vector is given by , 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
Given the curve
in , the unit tangent vector at is given by
If is given parametrically in the n-dimensional coordinate system xi (here we have used superscripts as an index instead of the usual subscript) by or
then the tangent vector field is given by
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.
Let be a differentiable function and let be a vector in . We define the directional derivative in the direction at a point by
The tangent vector at the point may then be defined as
Let be differentiable functions, let be tangent vectors in at , and let . Then
Tangent vector on manifolds
Let be a differentiable manifold and let be the algebra of real-valued differentiable functions on . Then the tangent vector to at a point in the manifold is given by the derivation which shall be linear — i.e., for any and we have
Note that the derivation will by definition have the Leibniz property