Tangent vector

 ( 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 $x$.

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

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Calculus Let $\backslash mathbf\{r\}(t)$ be a parametric smooth curve. The tangent vector is given by $\backslash mathbf\{r\}\text{'}(t)$ provided it exists and provided $\backslash mathbf\{r\}\text{'}(t)\backslash neq\; \backslash mathbf\{0\}$, where we have used a prime instead of the usual dot to indicate differentiation with respect to parameter .J. Stewart (2001) The unit tangent vector is given by $$\backslash mathbf\{T\}(t)\; =\; \backslash frac\{\backslash mathbf\{r\}\text{'}(t)\}$$\,.

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Example Given the curve $$\backslash mathbf\{r\}(t)\; =\; \backslash left\backslash \{\backslash left(1+t^2,\; e^\{2t\},\; \backslash cos\{t\}\backslash right)\; \backslash mid\; t\backslash in\backslash R\backslash right\backslash \}$$ in $\backslash R^3$, the unit tangent vector at $t\; =\; 0$ is given by $$\backslash mathbf\{T\}(0)\; =\; \backslash frac\{\backslash mathbf\{r\}\text{'}(0)\}\{\backslash |\backslash mathbf\{r\}\text{'}(0)\backslash |\}\; =\; \backslash left.\backslash frac\{(2t,\; 2e^\{2t\},\; -\backslash sin\{t\})\}\{\backslash sqrt\{4t^2\; +\; 4e^\{4t\}\; +\; \backslash sin^2\{t\}\}\}\backslash right|\_\{t=0\}\; =\; (0,1,0)\backslash ,.$$ Where the components of the tangent vector are found by taking the derivative of each corresponding component of the curve with respect to $t$.

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Contravariance If $\backslash mathbf\{r\}(t)$ is given parametrically in the n-dimensional coordinate system (here we have used superscripts as an index instead of the usual subscript) by $\backslash mathbf\{r\}(t)\; =\; (x^1(t),\; x^2(t),\; \backslash ldots,\; x^n(t))$ or $$\backslash mathbf\{r\}\; =\; x^i\; =\; x^i(t),\; \backslash quad\; a\backslash leq\; t\backslash leq\; b\backslash ,,$$ then the tangent vector field $\backslash mathbf\{T\}\; =\; T^i$ is given by $$T^i\; =\; \backslash frac\{dx^i\}\{dt\}\backslash ,.$$ Under a change of coordinates $$u^i\; =\; u^i(x^1,\; x^2,\; \backslash ldots,\; x^n),\; \backslash quad\; 1\backslash leq\; i\backslash leq\; n$$ the tangent vector $\backslash bar\{\backslash mathbf\{T\}\}\; =\; \backslash bar\{T\}^i$ in the -coordinate system is given by $$\backslash bar\{T\}^i\; =\; \backslash frac\{du^i\}\{dt\}\; =\; \backslash frac\{\backslash partial\; u^i\}\{\backslash partial\; x^s\}\; \backslash frac\{dx^s\}\{dt\}\; =\; T^s\; \backslash frac\{\backslash partial\; u^i\}\{\backslash partial\; x^s\}$$ 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.D. Kay (1988)

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Definition Let $f:\; \backslash R^n\; \backslash to\; \backslash R$ be a differentiable function and let $\backslash mathbf\{v\}$ be a vector in $\backslash R^n$. We define the directional derivative in the $\backslash mathbf\{v\}$ direction at a point $\backslash mathbf\{x\}\; \backslash in\; \backslash R^n$ by $$\backslash nabla\_\backslash mathbf\{v\}\; f(\backslash mathbf\{x\})\; =\; \backslash left.\backslash frac\{d\}\{dt\}\; f(\backslash mathbf\{x\}\; +\; t\backslash mathbf\{v\})\backslash right|\_\{t=0\}\; =\; \backslash sum\_\{i=1\}^\{n\}\; v\_i\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\_i\}(\backslash mathbf\{x\})\backslash ,.$$ The tangent vector at the point $\backslash mathbf\{x\}$ may then be definedA. Gray (1993) as $$\backslash mathbf\{v\}(f(\backslash mathbf\{x\}))\; \backslash equiv\; (\backslash nabla\_\backslash mathbf\{v\}(f))\; (\backslash mathbf\{x\})\backslash ,.$$

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Properties Let $f,g:\backslash mathbb\{R\}^n\backslash to\backslash mathbb\{R\}$ be differentiable functions, let $\backslash mathbf\{v\},\backslash mathbf\{w\}$ be tangent vectors in $\backslash mathbb\{R\}^n$ at $\backslash mathbf\{x\}\backslash in\backslash mathbb\{R\}^n$, and let $a,b\backslash in\backslash mathbb\{R\}$. Then

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

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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 $D\_v:A(M)\backslash rightarrow\backslash mathbb\{R\}$ which shall be linear — i.e., for any $f,g\backslash in\; A(M)$ and $a,b\backslash in\backslash mathbb\{R\}$ we have

$D\_v(af+bg)=aD\_v(f)+bD\_v(g)\backslash ,.$

Note that the derivation will by definition have the Leibniz property

$D\_v(f\backslash cdot\; g)(x)=D\_v(f)(x)\backslash cdot\; g(x)+f(x)\backslash cdot\; D\_v(g)(x)\backslash ,.$

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See also

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Bibliography

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Categories: Vectors

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