Tangent_vector Knowpia

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 edit

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

Calculus edit

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 $t$ .^[1] The unit tangent vector is given by

\mathbf {T} (t)={\frac {\mathbf {r} '(t)}{|\mathbf {r} '(t)|}}\,.

Example edit

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)\,.

Contravariance edit

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

u i

-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 edit

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^[3] as

\mathbf {v} (f(\mathbf {x} ))\equiv (\nabla _{\mathbf {v} }(f))(\mathbf {x} )\,.

Properties edit

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

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

Tangent vector on manifolds edit

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)\,.

References edit

^ J. Stewart (2001)
^ D. Kay (1988)
^ A. Gray (1993)

Bibliography edit

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.

[1] J. Stewart (2001)

[2] D. Kay (1988)

[3] A. Gray (1993)

[1]

[2]

[3]

Summary