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Today in a variational principles lecture we were shown the following:

If we have two points on the plane, namely, $(x_1,y_1)$ and $(x_2,y_2)$ and we have a curve $y(x)$ s.t. $y(x_1)=y_1$ and $y(x_2)=y_2$ then we can calculate the length of the curve by considering the arc length of the curve at a point $s(x)$ with $s(x_1)=0$.

We calculate the length of the curve by the integral:

$$ \int_{x_1}^{x_2}{\sqrt{1+(y'(x))^2}}dx $$

This result comes from considering small changes in $x$ and $y$ resulting in small changes in $s$.

Specifically, we look at $ds^2=dx^2+dy^2$. We were told this is because the curve can be approximated by a line when looked at closely. Is there a more rigorous justification to this or is it that simple?

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  • $\begingroup$ Look into the idea of a homeomorphism, that may help $\endgroup$ – JacobCheverie May 2 at 18:15
  • $\begingroup$ Anything specifically, or homeomorphisms in general? $\endgroup$ – Sam.S May 2 at 18:18
  • $\begingroup$ Wikipedia explains the justification for the definition of arc length for a curve $f : [a,b]\to\mathbb R^2$. For your case, the curve is $f(x) = (x, y(x))$. $\endgroup$ – Rahul May 2 at 18:19
  • $\begingroup$ I'm not sure if it'll even apply because I am not an expert, but in general, homeomorphisms rigorously treat the idea of a space deforming to another. It just made me think of your curve "deforming" to a line at a small enough scale. $\endgroup$ – JacobCheverie May 2 at 18:20
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The "rigor" comes from the fact that we literally define the arc length to be the (infinite) sum of the small straight line segments that "make up" the arbitrary arc.

It is very much like integrals. The concept of "area of an arbitrary region" itself doesn't have much meaning. So we do three things: 1) We define the area of a rectangle (our basic region). 2) We "cover" the region with infinitesimally small rectangles and sum their areas. 3) We call that number (result of the sum) the area of the arbitrary region.

Same goes for the arc length. 1) We define the length of a straight line segment (using the Pythagorean theorem). 2) We approximate/cover the arbitrary arc with infinitesimally small line segments. 3) We add the lengths of the small lines segments together and call the result, the length of the arc.

For more info about this approach of defining areas (and by extension arc lengths) take a look at chapter 1 (page 48) of Calculus, Volume 1.

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I think it is based on Pythagorean theorem and at any point on a curve, the slope at it is given by $\frac{dy}{dx}$. Let $ds$ be the length of a small arc on that point. So, it can be approximated to a small line. Thus we have, $dx$ as the base, $dy$ as the height and $ds$ as the hypotenuse.

So, $$(ds)^2 = (dx)^2 + (dy)^2$$

small arc

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  • $\begingroup$ I am aware the Pythagorean Theorem is used, but my confusion lies with a small arc on that point being approximated by a line. $\endgroup$ – Sam.S May 2 at 18:13
  • $\begingroup$ We can use the limiting value that for small $\theta$, $sin\theta \approx \theta$. Now $sin\theta$ is the ratio of two lengths,( e.g. consider $\frac{height}{hypotenuse}$) and $\theta$ is the ratio of of arc length and radius. So if we take the height of a small triangle equal to the radius of a small arc of circle having a small angle, we can approximately consider the hypotenuse as the length of the arc $\endgroup$ – Ak19 May 2 at 18:24
  • $\begingroup$ You may refer en.wikipedia.org/wiki/Small-angle_approximation for better understanding of the limits. $\endgroup$ – Ak19 May 2 at 18:25

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