Difference between revisions of "Curve"

Revision as of 19:18, 25 March 2015 (view source) Alec (Talk | contribs) (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...")		Latest revision as of 21:30, 28 March 2015 (view source) Alec (Talk | contribs) m
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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"
−	==Definitions==	+	==Level curve==
−	===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>
−	===Parameterisation===	+	==Parametrisation==
−	A parameterisation of a curve in {{M|\mathbb{R}^n}} is a function:	+	'''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====	+	===Linking with Level curves===
−	A parameterisation whos image is all (or a part of) a level curve is called a parameterisation (of part) of the level curve.	+	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}}

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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"

Level curve

Given a

$$f:\mathbb{R}^n\rightarrow\mathbb{R}$$ and a $c\in\mathbb{R}$ we define the level curve as follows^[1]:

$$\mathcal{C}=\{x\in\mathbb{R}^n|f(x)=c\}$$

A more useful notation is

$$\mathcal{C}_\alpha=\{x\in\mathbb{R}^n|f(x)=\alpha\}$$

Parametrisation

Note: see Parametrisation for details

A parametrisation of a curve in $\mathbb{R}^n$ is a function^[2]:

$$\gamma:(a,b)\rightarrow\mathbb{R}^n$$ with $$-\infty\le a< b\le +\infty$$

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 $\gamma$ are $\gamma(t)=(\gamma_1(t),\gamma_2(t),\cdots,\gamma_n(t))$

Differentiation

The derivative $\frac{d\gamma}{dt}=\dot{\gamma}(t)=(\frac{d\gamma_1}{dt},\frac{d\gamma_2}{dt},\cdots,\frac{d\gamma_n}{dt})$

See also

• Reparametrisation
• Parametrisation

References

1. Jump up ↑ Elementary Differential Geometry - Pressley - Springer SUMS series
2. Jump up ↑ Elementary Differential Geometry - Pressley - Springer SUMS series