Curve
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]
