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 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 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.