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. In other words, 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 the 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

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

in $\mathbb {R} ^{3}$ , the unit tangent vector at time $t=0$ 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.^[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

{\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

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

Tangent vector on manifolds

Let $M$ be a differentiable manifold and let $A(M)$ be the algebra of real-valued differentiable functions $M$ . 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)=D_{v}(f)\cdot g(x)+f(x)\cdot D_{v}(g)\,.

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 .

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