Difference between revisions of "Curve"
(Created page with "A curve can mean many things. It is reasonably standard to say however that a curve is any one dimensional "thing" ==Definitions== ===Level curve=== Given a <math>f:\mathbb{R...") |
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A curve can mean many things. It is reasonably standard to say however that a curve is any one dimensional "thing" | A curve can mean many things. It is reasonably standard to say however that a curve is any one dimensional "thing" | ||
| − | + | ==Level curve== | |
| − | + | Given a <math>f:\mathbb{R}^n\rightarrow\mathbb{R}</math> and a {{M|c\in\mathbb{R} }} we define the level curve as follows<ref> | |
| − | Given a <math>f:\mathbb{R}^n\rightarrow\mathbb{R}</math> and a {{M|c\in\mathbb{R} }} we define the level curve as follows: | + | Elementary Differential Geometry - Pressley - Springer SUMS series</ref>: |
<math>\mathcal{C}=\{x\in\mathbb{R}^n|f(x)=c\}</math> | <math>\mathcal{C}=\{x\in\mathbb{R}^n|f(x)=c\}</math> | ||
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A more useful notation is <math>\mathcal{C}_\alpha=\{x\in\mathbb{R}^n|f(x)=\alpha\}</math> | A more useful notation is <math>\mathcal{C}_\alpha=\{x\in\mathbb{R}^n|f(x)=\alpha\}</math> | ||
| − | == | + | ==Parametrisation== |
| − | A | + | '''Note:''' see [[Parametrisation]] for details |
| + | |||
| + | |||
| + | A parametrisation of a curve in {{M|\mathbb{R}^n}} is a function<ref> | ||
| + | Elementary Differential Geometry - Pressley - Springer SUMS series</ref>: | ||
<math>\gamma:(a,b)\rightarrow\mathbb{R}^n</math> with <math>-\infty\le a< b\le +\infty</math> | <math>\gamma:(a,b)\rightarrow\mathbb{R}^n</math> with <math>-\infty\le a< b\le +\infty</math> | ||
| − | + | ===Linking with Level curves=== | |
| − | A | + | A parametrisation whos image is all (or a part of) a level curve is called a parameterisation (of part) of the level curve. |
| + | |||
| + | ===Components=== | ||
| + | The component functions of {{M|\gamma}} are {{M|1=\gamma(t)=(\gamma_1(t),\gamma_2(t),\cdots,\gamma_n(t))}} | ||
| + | |||
| + | ===Differentiation=== | ||
| + | The derivative {{M|1=\frac{d\gamma}{dt}=\dot{\gamma}(t)=(\frac{d\gamma_1}{dt},\frac{d\gamma_2}{dt},\cdots,\frac{d\gamma_n}{dt})}} | ||
| + | |||
| + | ==See also== | ||
| + | * [[Reparametrisation]] | ||
| + | * [[Parametrisation]] | ||
| + | ==References== | ||
| + | <references /> | ||
{{Definition|Differential Geometry|Geometry of Curves and Surfaces}} | {{Definition|Differential Geometry|Geometry of Curves and Surfaces}} | ||
Latest revision as of 21:30, 28 March 2015
A curve can mean many things. It is reasonably standard to say however that a curve is any one dimensional "thing"
Contents
Level curve
Given a [math]f:\mathbb{R}^n\rightarrow\mathbb{R}[/math] and a [ilmath]c\in\mathbb{R} [/ilmath] we define the level curve as follows[1]:
[math]\mathcal{C}=\{x\in\mathbb{R}^n|f(x)=c\}[/math]
A more useful notation is [math]\mathcal{C}_\alpha=\{x\in\mathbb{R}^n|f(x)=\alpha\}[/math]
Parametrisation
Note: see Parametrisation for details
A parametrisation of a curve in [ilmath]\mathbb{R}^n[/ilmath] is a function[2]:
[math]\gamma:(a,b)\rightarrow\mathbb{R}^n[/math] with [math]-\infty\le a< b\le +\infty[/math]
Linking with Level curves
A parametrisation whos image is all (or a part of) a level curve is called a parameterisation (of part) of the level curve.
Components
The component functions of [ilmath]\gamma[/ilmath] are [ilmath]\gamma(t)=(\gamma_1(t),\gamma_2(t),\cdots,\gamma_n(t))[/ilmath]
Differentiation
The derivative [ilmath]\frac{d\gamma}{dt}=\dot{\gamma}(t)=(\frac{d\gamma_1}{dt},\frac{d\gamma_2}{dt},\cdots,\frac{d\gamma_n}{dt})[/ilmath]
