Difference between revisions of "Parametrisation"
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| − | 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
Contents
[hide]Definition
A parametrisation γ is a function[1]:
γ:(a,b)→Rn with −∞≤a<b≤+∞
Often t is the parameter, so we talk of γ(t0) or γ(t)
Differentiation
TODO: Add picture
Intuitively we see that the gradient at t of γ is ≈γ(t+δt)−γ(t)δt taking the limit of δt→0 we get dγdt=limδt→0(γ(t+δt)−γ(t)δt) as usual.
Other notations for this include ˙γ
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 γ is ∥˙γ(t)∥
See also
References
- Jump up ↑ Elementary Differential Geometry - Pressley - Springer SUMS
