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 $c\in\mathbb{R}$ we define the level curve as follows:

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

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:

$$\gamma:(a,b)\rightarrow\mathbb{R}^n$$ 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