Tangent vector

For a more general — but much more technical — treatment of tangent vectors, see tangent space.

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 $x$.

Motivation

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 $\mathbf {r} ^{\prime }(t)$, 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

$\mathbf {T} (t)={\frac {\mathbf {r} ^{\prime }(t)}{|\mathbf {r} ^{\prime }(t)|}}\,.$

Example

Given the curve

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

in $\mathbb {R} ^{3}$, the unit tangent vector at $t=0$ is given by

$\mathbf {T} (0)={\frac {\mathbf {r} ^{\prime }(0)}{\|\mathbf {r} ^{\prime }(0)\|}}=\left.{\frac {(2t,2e^{2t},\ -\sin {t})}{\sqrt {4t^{2}+4e^{4t}+\sin ^{2}{t}}}}\right|_{t=0}=(0,1,0)\,.$

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.

Definition

Let $f:\mathbb {R} ^{n}\rightarrow \mathbb {R}$ 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

$D_{\mathbf {v} }f(\mathbf {x} )=\left.{\frac {d}{dt}}f(\mathbf {x} +t\mathbf {v} )\right|_{t=0}=\sum _{i=1}^{n}v_{i}{\frac {\partial f}{\partial x_{i}}}(\mathbf {x} )\,.$

The tangent vector at the point $\mathbf {x}$ may then be defined as

$\mathbf {v} (f(\mathbf {x} ))\equiv (D_{\mathbf {v} }(f))(\mathbf {x} )\,.$

Properties

Let $f,g:\mathbb {R} ^{n}\rightarrow \mathbb {R}$ 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

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

Tangent 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

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

Bibliography

• Gray, Alfred (1993), Modern Differential Geometry of Curves and Surfaces, Boca Raton: CRC Press.
• Stewart, James (2001), Calculus: Concepts and Contexts, Australia: Thomson/Brooks/Cole.
• Kay, David (1988), Schaums Outline of Theory and Problems of Tensor Calculus, New York: McGraw-Hill.