Wikipedia:Reference desk/Archives/Mathematics/2021 February 19

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February 19

Gabriel's Horn

Gabriel's Horn gives integrals for the horn's volume and surface area. The volume is done by chopping the horn into discs and calculating the volume of each disc pi*(1/x)^2*dx. That makes perfect sense to me. So why is the surface area so much more complicated? Why can't that be done by chopping into the same disks and calculating their edge area as 2*pi*(1/x)*dx? This gives the same result without involving the 1 + 1/x^4. -- SGBailey (talk) 22:59, 19 February 2021 (UTC)

Note: Gabriel's horn recent appeared in this Numberphile video. They did gloss over this point a bit there. The short answer is that the surface area integral has a ds factor while the volume integral has a dx factor, and since ds has a complicated expression in terms of f(x), surface area integrals tend to be more complicated. The root of the problem is that when you approximate the solid by a large number of cylinders, the difference in volume between the solid and it's approximation tends to 0 as the cylinders become thinner. But with surface area, you get a staircase effect which does not disappear as the slices get thinner. This is similar to the problem with approximating a diagonal line with a series of horizontal and vertical segments. The actual length of the line is given by √(∆x²+∆y²) where ∆x is the horizontal distance and ∆y is the vertical distance. But the length of the approximation by horizontal and vertical segments is ∆x+∆y, and this stays the same no matter how small the segments are. This is why for arclength you need to approximate to the curve by secant lines, not just any line segements that happen to stay near the curve. Part of the problem is that the philosophical basis for what it means for a curve to have a certain length, or for a curved surface to have a certain area are rather vague, at least when it comes to the way they teach it in undergraduate calculus. What usually happens (in my experience) is that the length is simply defined as the integral of ds, thus short circuiting the need for philosophical discussion. --RDBury (talk) 00:28, 20 February 2021 (UTC)

@User:RDBury What is the philosophical definition of length of a curve, if not integral of ds?- Abdul Muhsy talk 14:55, 20 February 2021 (UTC)

Since $ds$ stands for the differential of $s$, which stands for the arc length along the curve, it presupposes an arc length and so should not be used to define it. Moreover, it requires the curve to be differentiable. A definition that works for all (rectifiable) curves, also when they are not differentiable, is as follows. Let $(M,d)$ be a metric space, and let a curve in that space be given in the form of a continuous function $\gamma {:}\,[a,b]\to M,$ where $[a,b]$ is an interval of $\mathbb {R}$. A partition of $[a,b]$ is a strictly ascending sequence $P=(x_{0},x_{1},...,x_{n})$ such that $[a,b]=[x_{0},x_{n}].$ Define $\textstyle {L(P)=\sum _{i=0}^{n{-}1}d(\gamma (x_{i}),\gamma (x_{i{+}1}))}.$ The length (if it exists) along the curve from $\gamma (a)$ to $\gamma (b)$ is then given by $\textstyle {\sup _{P}L(P)},$ the supremum over all partitions. This may be infinite. See How Long Is the Coast of Britain?  --Lambiam 22:47, 20 February 2021 (UTC)

This is a more general definition but I still don't see a philosophical basis for it; it basically gives a method of computation and defines the length to be the result of the computation. There is a perhaps apocryphal story that when Alfred Binet (the inventor of the first IQ test) was asked to define intelligence, he said "It's what my test measures." Perhaps it's too subtle a point to settle here, but I feel that the usual definitions suffer from the same kind of circularity: "This is the method to compute X."/"What is X?"/"X is what the method computes." From a mathematical point of view this is fine since a mathematical definition can be anything; mathematics does not require that the definition matches any preconceived ideas about you're defining, but if you are claiming such a match then there should be some basis for the claim. Instead of going more general and abstract it makes sense from a philosophical point of view to look at what it would mean in real life. I have an irregular object and I want to know the area of its surface. A practical method might be to paint it with a layer of paint d units thick, then the area is approximately the volume of paint used divided by d. I might claim that the approximation gets more accurate when d becomes smaller, and so define the area as the limit of the approximation as d approaches 0. This is another case of method of computation used as a definition, but this time the method of computation is based on the real life knowledge that a certain volume of paint covers an area to a given thickness. For the length of a curve in the plain you might try to imagine 2-dimensional paint, but perhaps a better way would be imagine the curve being approximated by a chain. We know that real life chains have a measurable length and if the chain is made to approximate a curve in some reasonable way then we can say that the length of the curve is approximately the length of the chain. This idea could be turned into a method of computation, say by idealizing the chain as a collection of line segments, the links, and postulating that the accuracy of the measurement increases as the length of the links gets smaller and the distance they are allowed to stray from the curve becomes smaller. The result would be very similar to the definition you gave above, but I think it would take a bit of work to prove that the two versions actually produce the same result. --RDBury (talk) 00:47, 21 February 2021 (UTC)

Thanks, but I don't really see any difference between the two definitions. The philosophy to me seems the same, "secant lines approximate the arc" and hence "(approx) arc length = (total) secant lines length". The fact that one of the definitions is based on real life doesn't necessarily confer it with some deeper truth. However it does make it easier to understand, and is appreciated. - Abdul Muhsy talk 04:57, 21 February 2021 (UTC)

The definition I gave above is essentially the same as that given in Curve § Length of a curve. It is indeed a bit clunky. The idea of using paint (in mathematical terminology, basically dilation with a small closed ball) and (polygonal) chains suggested another approach to me, which I find more elegant. Given a metric space $(M,d)$, the metric induces another metric, known as the Hausdorff metric, on the set of non-empty subspaces of $M$. We can use this to define the distance $d(\gamma ,\gamma ')$ between two curves in $M$ as the Hausdorff distance between their images in $M$. Denoting the length of polyline $\pi$ by $L(\pi )$, we then extend $L$ to curves in general by:

$L(\gamma )=\lim _{\epsilon \downarrow 0}~\inf _{d(\gamma ,\pi )\leq \epsilon }L(\pi ).$

 --Lambiam 06:28, 22 February 2021 (UTC)

An interesting example where one can readily see that summing edge areas of thin disk slices doesn't work is for the area of a sphere, given by the well-known formula $4\pi r^{2}.$ Pick a great circle, which we'll call "the equator". Consider a cylinder tightly enclosing the sphere, with radius $r$ and height $2r,$ touching it along the equator. Slice both orthogonal to the cylinder axis. Clearly, the edge areas of the slices of the sphere are everywhere less than those of the corresponding slices of the cylinder, except at the equator. Yet the surface area of the cylinder (not counting top and bottom but only the curved part) is $2\pi rh,$ where $h=2r,$ resulting in $4\pi r^{2},$ the same as for the sphere. In general, projecting sphere regions from the cylinder axis onto the cylinder is an equal-area map projection, the Lambert cylindrical equal-area projection.  --Lambiam 00:57, 20 February 2021 (UTC)