Variations of Regular Curves problem

[MxwllsPersuasns]

Homework Statement

Let γ_s : I → Rⁿ, s ∈ (−δ, δ), > 0, be a variation with compact support K ⊂ I' of a regular curve γ = γ₀. Show that there exists some 0 < δ ≤ ε such that γ_s is a regular curve for all s ∈ (−δ, δ). Thus, we may assume w.l.o.g. that any variation of a regular curve consists of regular curves.

Homework Equations

The Attempt at a Solution

*I didn't want to phrase my title exactly that way but I'm a bit desperate since I post here a lot and always put a lot of effort and care into formulating my questions and while I may ask longer questions I always partition it up nicely and emphasize an a la carte approach to answering them so as to not be overwhelmed, but don't seem to get too much traffic (or at least help) very often (though when I do it is extraordinary) and just wanted to emphasize I really need help here*

So my main reason for posting this is that (due to extenuating circumstances) I missed a week of lectures and I emailed my professor to explain some of the terminology or perhaps provide me with references or sources for the material and his response was essentially that the terminology and everything we were using was so specific no one resource would have it and to get notes from a classmate. However I don't know any of my classmates and so here I am, hoping you intelligent people can help explain some of this to me. So...

I guess I understand what the question is asking; to show that for some value, δ, of the parameter s (what exactly is this parameter? Switches through the "variations" on the curve?) the 'variations' of a regular curve are regular curves.

i) I'm not sure what is meant by "variations" on the curves.. it evokes images of the calculus of variations, but as I haven't studied much of that I am unfamiliar...

ii) What does having "compact support" mean? It looks like K is a subset of the interval I. The term 'compact' evokes thoughts of "compactness" from point-set topology, which I know is related to the closure of a set (or interval) so perhaps this means that for the closed interval K ⊂ I?

iii) What should I do to go about proving this? There's a hint associated with the statement that basically says I should; understand that since γ is regular that its derivative is never 0 (||γ'(t)|| ≥ M(t) > 0) and also by continuity of γ_s in s means then that we know that there exists some δ(t) > 0 such that ||γ'_s(t)|| ≥ (1/2)M(t) > 0 for s ∈ (−δ, δ). The idea then is to take the min of all those δ(t) as t ranges over I. "We will need the fact that by the compact support of the variation γ_s you only have to show regularity of γ_s for t ∈ K.

 

[lippyka]

I'm sorry my question may be a bit long, and I know this is a lot to ask but I really don't know what to do or how to approach this or even just interpret the hints given. I'm sorry if I'm being a bit vague but I hope someone can help!