# Curve

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 [ilmath]c\in\mathbb{R} [/ilmath] 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 [ilmath]\mathbb{R}^n[/ilmath] is a function[2]:

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

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]