Editing Tangent vector (section)

=== Contravariance ===
If <math>\mathbf{r}(t)</math> is given parametrically in the [[n-dimensional coordinate system|''n''-dimensional coordinate system]] {{math|''x<sup>i</sup>''}} (here we have used superscripts as an index instead of the usual subscript) by <math>\mathbf{r}(t) = (x^1(t), x^2(t), \ldots, x^n(t))</math> or
<math display="block">\mathbf{r} = x^i = x^i(t), \quad a\leq t\leq b\,,</math>
then the tangent vector field <math>\mathbf{T} = T^i</math> is given by
<math display="block">T^i = \frac{dx^i}{dt}\,.</math>
Under a change of coordinates
<math display="block">u^i = u^i(x^1, x^2, \ldots, x^n), \quad 1\leq i\leq n</math>
the tangent vector <math>\bar{\mathbf{T}} = \bar{T}^i</math> in the {{math|''u<sup>i</sup>''}}-coordinate system is given by
<math display="block">\bar{T}^i = \frac{du^i}{dt} = \frac{\partial u^i}{\partial x^s} \frac{dx^s}{dt} = T^s \frac{\partial u^i}{\partial x^s}</math>
where we have used the [[Einstein notation|Einstein summation convention]]. Therefore, a tangent vector of a smooth curve will transform as a [[Covariance and contravariance of vectors|contravariant]] tensor of order one under a change of coordinates.<ref>D. Kay (1988)</ref>