tangent vector

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

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 {{mvar|t}}.J. Stewart (2001) The unit tangent vector is given by

$$\backslash mathbf\{T\}(t)\; =\; \backslash frac\{\backslash mathbf\{r\}\text{'}(t)\}$$

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

$$\backslash mathbf\{T\}(0)\; =\; \backslash frac\{\backslash mathbf\{r\}\text{'}(0)\}\{\backslash |\backslash mathbf\{r\}\text{'}(0)\backslash |\}\; =\; \backslash left.\backslash frac\{(2t,\; 2e^\{2t\},\; -\backslash sin\{t\})\}\{\backslash sqrt\{4t^2\; +\; 4e^\{4t\}\; +\; \backslash sin^2\{t\}\}\}\backslash right|\_\{t=0\}\; =\; (0,1,0)\backslash ,.$$

If $\backslash mathbf\{r\}(t)$ is given parametrically in the n-dimensional coordinate system {{math|xⁱ}} (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 {{math|uⁱ}}-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)

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

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

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. $\backslash mathbf\{v\}(fg)=f(\backslash mathbf\{x\})\backslash mathbf\{v\}(g)+g(\backslash mathbf\{x\})\backslash mathbf\{v\}(f)\backslash ,.$

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

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

• {{slink|Differentiable curve#Tangent vector}}
• {{slink|Differentiable surface#Tangent plane and normal vector}}

• {{citation|first=Alfred|last=Gray|title=Modern Differential Geometry of Curves and Surfaces|publisher=CRC Press|publication-place=Boca Raton|year=1993}}.
• {{citation|first=James|last=Stewart|title=Calculus: Concepts and Contexts|publisher=Thomson/Brooks/Cole|publication-place=Australia|year=2001}}.
• {{citation|first=David|last=Kay|title=Schaums Outline of Theory and Problems of Tensor Calculus|publisher=McGraw-Hill|publication-place=New York|year=1988}}.

Category:Vectors (mathematics and physics)