Vector tangent to a curve or surface at a given point

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 **R**^{n}. 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 $x$ is a linear derivation of the algebra defined by the set of germs at $x$.

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Motivation

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

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Calculus

Let $\mathbf {r} (t)$ be a parametric smooth curve. The tangent vector is given by $\mathbf {r} '(t)$, 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

$\mathbf {T} (t)={\frac {\mathbf {r} '(t)}{|\mathbf {r} '(t)|))\,.$

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Example

Given the curve

$\mathbf {r} (t)=\left\{\left(1+t^{2},e^{2t},\cos {t}\right)\mid t\in \mathbb {R} \right\))$

in $\mathbb {R} ^{3))$, the unit tangent vector at $t=0$ is given by
$\mathbf {T} (0)={\frac {\mathbf {r} '(0)}{\|\mathbf {r} '(0)\|))=\left.{\frac {(2t,2e^{2t},-\sin {t})}{\sqrt {4t^{2}+4e^{4t}+\sin ^{2}{t))))\right|_{t=0}=(0,1,0)\,.$

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Contravariance

If $\mathbf {r} (t)$ is given parametrically in the *n*-dimensional coordinate system *x*^{i} (here we have used superscripts as an index instead of the usual subscript) by $\mathbf {r} (t)=(x^{1}(t),x^{2}(t),\ldots ,x^{n}(t))$ or

$\mathbf {r} =x^{i}=x^{i}(t),\quad a\leq t\leq b\,,$

then the tangent vector field $\mathbf {T} =T^{i))$ is given by
$T^{i}={\frac {dx^{i)){dt))\,.$

Under a change of coordinates
$u^{i}=u^{i}(x^{1},x^{2},\ldots ,x^{n}),\quad 1\leq i\leq n$

the tangent vector ${\bar {\mathbf {T} ))={\bar {T))^{i))$ in the *u*^{i}-coordinate system is given by
${\bar {T))^{i}={\frac {du^{i)){dt))={\frac {\partial u^{i)){\partial x^{s))}{\frac {dx^{s)){dt))=T^{s}{\frac {\partial u^{i)){\partial x^{s))))$

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.^{[2]}