Parametrisation

Definition

A parametrisation $\gamma$ is a function:

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

Often $t$ is the parameter, so we talk of $\gamma(t_0)$ or $\gamma(t)$

Differentiation

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Intuitively we see that the gradient at $t$ of $\gamma$ is $\approx\frac{\gamma(t+\delta t)-\gamma(t)}{\delta t}$ taking the limit of $\delta t\rightarrow 0$ we get $\frac{d\gamma}{dt}=\lim_{\delta t\rightarrow 0}(\frac{\gamma(t+\delta t)-\gamma(t)}{\delta t})$ as usual.

Other notations for this include $\dot{\gamma}$

Speed

Speed is the rate of change of distance (velocity is the rate of change of position - which are both vector quantities) - from differentiating the Arc length we define speed as:

The speed at $t$ of $\gamma$ is $$\|\dot{\gamma}(t)\|$$

See also

• Unit speed parametrisation
• Arc length
• Curve
• Reparametrisation
• Smooth