Difference between revisions of "Parametrisation"
m |
m (→Differentiation) |
||
| Line 9: | Line 9: | ||
==Differentiation== | ==Differentiation== | ||
{{Todo|Add picture}} | {{Todo|Add picture}} | ||
| − | Intuitively we see that the gradient at {{M|t}} of {{M|\gamma}} is < | + | Intuitively we see that the gradient at {{M|t}} of {{M|\gamma}} is <m>\approx\frac{\gamma(t+\delta t)-\gamma(t)}{\delta t}</m> taking the limit of {{M|\delta t\rightarrow 0}} we get {{M|1=\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 {{M|\dot{\gamma} }} | Other notations for this include {{M|\dot{\gamma} }} | ||
Latest revision as of 11:10, 12 June 2015
Definition
A parametrisation [ilmath]\gamma[/ilmath] is a function[1]:
[math]\gamma:(a,b)\rightarrow\mathbb{R}^n[/math] with [math]-\infty\le a< b\le +\infty[/math]
Often [ilmath]t[/ilmath] is the parameter, so we talk of [ilmath]\gamma(t_0)[/ilmath] or [ilmath]\gamma(t)[/ilmath]
Differentiation
TODO: Add picture
Intuitively we see that the gradient at [ilmath]t[/ilmath] of [ilmath]\gamma[/ilmath] is [ilmath]\approx\frac{\gamma(t+\delta t)-\gamma(t)}{\delta t}[/ilmath] taking the limit of [ilmath]\delta t\rightarrow 0[/ilmath] we get [ilmath]\frac{d\gamma}{dt}=\lim_{\delta t\rightarrow 0}(\frac{\gamma(t+\delta t)-\gamma(t)}{\delta t})[/ilmath] as usual.
Other notations for this include [ilmath]\dot{\gamma} [/ilmath]
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 [ilmath]t[/ilmath] of [ilmath]\gamma[/ilmath] is [math]\|\dot{\gamma}(t)\|[/math]
See also
References
- ↑ Elementary Differential Geometry - Pressley - Springer SUMS
