Variations of Regular Curves problem

In summary, the problem at hand is to show that for a variation with compact support of a regular curve, there exists a value of δ such that the variation is also a regular curve. The term "variation" is not explicitly defined, but it is suggested that it may be related to the calculus of variations. The phrase "compact support" indicates that the subset K of the interval I is closed. To prove this, it is suggested to use the fact that the derivative of a regular curve is never 0 and the continuity of the variation in s. The goal is to find the minimum of all δ values for s in (-δ, δ) as t ranges over I. The hint also states that it is only necessary to show regular
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Homework Statement



Let γs : 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 γs 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 γs 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 γs you only have to show regularity of γs for t ∈ K.
 
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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!
 

Related to Variations of Regular Curves problem

1. What is the Variations of Regular Curves problem?

The Variations of Regular Curves problem is a mathematical problem that involves finding the maximum or minimum values of a function that represents a regular curve. It is often used in optimization and calculus courses.

2. What are some examples of regular curves?

Some examples of regular curves include circles, ellipses, parabolas, and hyperbolas. These curves have a smooth and consistent shape.

3. How is the Variations of Regular Curves problem solved?

The problem is typically solved by using calculus techniques, such as finding the derivative of the function and setting it equal to zero to find the critical points. These critical points can then be used to determine the maximum or minimum values of the function.

4. What real-world applications use the Variations of Regular Curves problem?

The Variations of Regular Curves problem has many real-world applications, such as in engineering, physics, and economics. For example, it can be used to optimize the design of a bridge or to find the most efficient way to allocate resources.

5. Are there any variations of the Variations of Regular Curves problem?

Yes, there are variations of the problem that involve finding the maximum or minimum values of a function over a specific interval or with certain constraints. These variations are commonly used in advanced calculus and optimization courses.

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