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Tangent vector

Vector tangent to a curve or surface at a given point

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Source updated October 29, 2025

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 $x$ is a linear derivation of the algebra defined by the set of germs at $x$ .

01Motivation

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

Calculus

Let $\mathbf {r} (t)$ be a parametric smooth curve. The tangent vector is given by $\mathbf {r} '(t)$ provided it exists and provided $\mathbf {r} '(t)\neq \mathbf {0}$ , 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 $\mathbf {T} (t)={\frac {\mathbf {r} '(t)}{|\mathbf {r} '(t)|}}\,.$

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 $\mathbf {T} (0)={\frac {\mathbf {r} '(0)}{\|\mathbf {r} '(0)\|}}=\left.{\frac {(2t,2e^{2t},-\sin {t})}{\sqrt {4t^{2}+4e^{4t}+\sin ^{2}{t}}}}\right|_{t=0}=(0,1,0)\,.$ Where the components of the tangent vector are found by taking the derivative of each corresponding component of the curve with respect to $t$ .

Contravariance

If $\mathbf {r} (t)$ is given parametrically in the n-dimensional coordinate system xⁱ (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 uⁱ-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.

02Definition

Let $f:\mathbb {R} ^{n}\to \mathbb {R}$ be a differentiable function and let $\mathbf {v}$ be a vector in $\mathbb {R} ^{n}$ . We define the directional derivative in the $\mathbf {v}$ direction at a point $\mathbf {x} \in \mathbb {R} ^{n}$ by $\nabla _{\mathbf {v} }f(\mathbf {x} )=\left.{\frac {d}{dt}}f(\mathbf {x} +t\mathbf {v} )\right|_{t=0}=\sum _{i=1}^{n}v_{i}{\frac {\partial f}{\partial x_{i}}}(\mathbf {x} )\,.$ The tangent vector at the point $\mathbf {x}$ may then be defined as $\mathbf {v} (f(\mathbf {x} ))\equiv (\nabla _{\mathbf {v} }(f))(\mathbf {x} )\,.$

03Properties

Let $f,g:\mathbb {R} ^{n}\to \mathbb {R}$ be differentiable functions, let $\mathbf {v} ,\mathbf {w}$ be tangent vectors in $\mathbb {R} ^{n}$ at $\mathbf {x} \in \mathbb {R} ^{n}$ , and let $a,b\in \mathbb {R}$ . Then

$(a\mathbf {v} +b\mathbf {w} )(f)=a\mathbf {v} (f)+b\mathbf {w} (f)$
$\mathbf {v} (af+bg)=a\mathbf {v} (f)+b\mathbf {v} (g)$
$\mathbf {v} (fg)=f(\mathbf {x} )\mathbf {v} (g)+g(\mathbf {x} )\mathbf {v} (f)\,.$

04Tangent vector on manifolds

Let $M$ be a differentiable manifold and let $A(M)$ be the algebra of real-valued differentiable functions on $M$ . Then the tangent vector to $M$ at a point $x$ in the manifold is given by the derivation $D_{v}:A(M)\rightarrow \mathbb {R}$ which shall be linear , i.e., for any $f,g\in A(M)$ and $a,b\in \mathbb {R}$ we have

$D_{v}(af+bg)=aD_{v}(f)+bD_{v}(g)\,.$

Note that the derivation will by definition have the Leibniz property

$D_{v}(f\cdot g)(x)=D_{v}(f)(x)\cdot g(x)+f(x)\cdot D_{v}(g)(x)\,.$

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Sources and credits

This article is adapted from the Wikipedia article “Tangent vector”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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