### - Art Gallery -

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 {\displaystyle x} 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.[1] The unit tangent vector is given by

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

Example

Given the curve

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

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

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

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.[2]

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[3] as
$${\displaystyle \mathbf {v} (f(\mathbf {x} ))\equiv (D_{\mathbf {v} }(f))(\mathbf {x} )\,.}$$

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 x ∈ $${\mathbf {x}}\in {\mathbb {R}}^{n}$$ , and let $$a,b\in {\mathbb {R}}$$. Then

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

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 {\displaystyle x} 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 ∈ A ( M ) {\displaystyle f,g\in A(M)} f,g\in A(M) and a , b ∈ R {\displaystyle a,b\in \mathbb {R} } a,b\in {\mathbb {R}} we have

Note that the derivation will by definition have the Leibniz property

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

References

J. Stewart (2001)
D. Kay (1988)

A. Gray (1993)

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.

Mathematics Encyclopedia

World

Index