All questions with tag [math: arc-length]

1

Examples of functions whose arc-length from the origin is given by their derivative

I'm seeking features $y:\mathbb{R}\rightarrow\mathbb{R}$ such that $$\int_{0}^{a} \sqrt{1+\left(\frac{dy}{dx}\right)^{2}} dx = \frac{dy}{dx}\Bigg|_{a}$$ (this sort of seems like a calculus - of - variants type trouble, yet I do not have any kind of experience with the calculus of variations)
2022-07-10 05:00:10
1

Can anyone tell me why the arclength integral is a lower semicontinuous function on the set of continuously differentiable real-valued functions?

I uploaded the question stating that it was upper semicontinuous, yet that was most definitely incorrect. I am attempting to confirm lower semicontinuity.
2022-06-12 10:28:56
2

Can anyone tell me why the arclength integral is an uppersemicontinuous function on the set of continuously differentiable real-valued functions?

The continually differentiable features are outfitted with the geography generated by the sup standard. I recognize that I can make the arclength indispensable close to the arclength of a piecewise straight function. My suggestion is to take 2 features which are appropriately close with each other, and also take 2 piecewise straight features whi...
2022-06-10 21:26:11