Editing Tangent vector (section)

== Motivation ==
Before proceeding to a general definition of the tangent vector, we discuss its use in [[calculus]] and its [[tensor]] properties.

=== Calculus ===
Let <math>\mathbf{r}(t)</math> be a parametric [[smooth curve]]. The tangent vector is given by <math>\mathbf{r}'(t)</math> provided it exists and provided <math>\mathbf{r}'(t)\neq \mathbf{0}</math>, where we have used a prime instead of the usual dot to indicate differentiation with respect to parameter {{mvar|t}}.<ref>J. Stewart (2001)</ref> The unit tangent vector is given by
<math display="block">\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}\,.</math>

==== Example ====
Given the curve
<math display="block">\mathbf{r}(t) = \left\{\left(1+t^2, e^{2t}, \cos{t}\right) \mid t\in\R\right\}</math>
in <math>\R^3</math>, the unit tangent vector at <math>t = 0</math> is given by
<math display="block">\mathbf{T}(0) = \frac{\mathbf{r}'(0)}{\|\mathbf{r}'(0)\|} = \left.\frac{(2t, 2e^{2t}, -\sin{t})}{\sqrt{4t^2 + 4e^{4t} + \sin^2{t}}}\right|_{t=0} = (0,1,0)\,.</math>

=== 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>