tangent vectors

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 $x$.

Motivation

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

Calculus

Let $\mathbf(t)$ be a parametric smooth curve. The tangent vector is given by $\mathbf'(t)$, where we have used a prime instead of the usual dot to indicate differentiation with respect to parameter . The unit tangent vector is given by
$\mathbf(t) = \frac\,.$

Example

Given the curve
$\mathbf(t) = \left\$
in $\R^3$, the unit tangent vector at $t = 0$ is given by
$\mathbf(0) = \frac = \left.\frac\_ = (0,1,0)\,.$

Contravariance

If $\mathbf(t)$ is given parametrically in the ''n''-dimensional coordinate system (here we have used superscripts as an index instead of the usual subscript) by $\mathbf(t) = (x^1(t), x^2(t), \ldots, x^n(t))$ or
$\mathbf = x^i = x^i(t), \quad a\leq t\leq b\,,$
then the tangent vector field $\mathbf = T^i$ is given by
$T^i = \frac\,.$
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 $\bar = \bar^i$ in the -coordinate system is given by
$\bar^i = \frac = \frac \frac = T^s \frac$
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 $f: \R^n \to \R$ be a differentiable function and let $\mathbf$ be a vector in $\R^n$. We define the directional derivative in the $\mathbf$ direction at a point $\mathbf \in \R^n$ by
$\nabla_\mathbf f(\mathbf) = \left.\frac f(\mathbf + t\mathbf)\_ = \sum_^ v_i \frac(\mathbf)\,.$
The tangent vector at the point $\mathbf$ may then be definedA. Gray (1993) as
$\mathbf(f(\mathbf)) \equiv (\nabla_\mathbf(f)) (\mathbf)\,.$

Properties

Let $f,g:\mathbb^n\to\mathbb$ be differentiable functions, let $\mathbf,\mathbf$ be tangent vectors in $\mathbb^n$ at $\mathbf\in\mathbb^n$, and let $a,b\in\mathbb$. Then
#$(a\mathbf+b\mathbf)(f)=a\mathbf(f)+b\mathbf(f)$
#$\mathbf(af+bg)=a\mathbf(f)+b\mathbf(g)$
#$\mathbf(fg)=f(\mathbf)\mathbf(g)+g(\mathbf)\mathbf(f)\,.$

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 $D_v:A(M)\rightarrow\mathbb$ which shall be linear — i.e., for any $f,g\in A(M)$ and $a,b\in\mathbb$ we have
:$D_v(af+bg)=aD_v(f)+bD_v(g)\,.$
Note that the derivation will by definition have the Leibniz property
:$D_v(f\cdot g)(x)=D_v(f)(x)\cdot g(x)+f(x)\cdot D_v(g)(x)\,.$

See also

*
*

References

Bibliography

* .
* .
* {{citation, first=David, last=Kay, title=Schaums Outline of Theory and Problems of Tensor Calculus, publisher=McGraw-Hill, publication-place=New York, year=1988.

Vectors (mathematics and physics)