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Can we take many $1D$ shapes with the shape of the letter v, make them stand one next to the other, and make a straight line out of it? It seems there is a kind of "height" (which the straight line lacks of) but it also seems that this height can become arbitrarily small if we multiply the number of v's (towards infinity).

This question showed up to me while watching a visual proof of Stokes' theorem. After seeing that, locally, $$ \iint_S \text{curl}\; {\bf G} \cdot d{\bf S} = \oint_C {\bf G} \cdot d{\bf r} $$

where $S$ and $C$ are taken to be infinitesimal surface and contour with a somewhat crazy shape ; they generalize the relation to a case where $S$ and $C$ are non infinitesimal surface and coutour (i.e. a random surface and a random contour) by saying we can divide a macroscopic surface into many infinitesimal surfaces (and the same applies to the corresponding contours).

My question then is, can we for example divide a disk into these crazy surfaces? Can we approximate the circle around the disk by the sum of paths such as the yellow path below?

By crazy surfaces and coutours I mean they took for the surfaces three faces (orthogonal each with another) of a cube (the red, green and blue in the image below), and for the contour, they took the yellow path, as you can see in the image below:enter image description here

PS: I have already understood that when we divide a contour into smaller contours, the circulation in the interior segments cancels out. The same for surfaces.

EDIT -

I am actually trying to check the math validity of this visual proof. After reading the Wiki article on Smooth Infinitesimal Analysis, they say "lines are made out of infinitesimally small segments" (in this theory). These segments can be thought of as being long enough to have a definite direction, but not long enough to be curved.

Should I draw the conclusion that even using Smooth Infinitesimal Analysis, a straight line can't be approximated by infinitesimally small v-s? Hence, that the visual proof of Stokes theorem is wrong?

Many thanks.

niobium
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    While you can talk about infinitesimals in some non-standard analysis, they really don't exist in standard analysis or calculus. Don't try to apply intuition to the concept, it will fail. – Thomas Andrews Dec 08 '24 at 21:30
  • Didn't get the chance to respond - you should have ignored the downvoters on Physics SE a while longer, maybe someone would have posted something insightful. I'm not sure if this will help, but the surface considered doesn't have to be planar - it can for example be a spherical cap, or otherwise have "hills" and "valleys" in it, and the contour is just the boundary. In the image, the surface considered has "creases", and looks like 3 faces of a cube, and the yellow contour is just the boundary in orthogonal projection - the six axis-aligned line segments of the boundary at 90° to each other. – Filip Milovanović Dec 09 '24 at 14:58
  • @FilipMilovanović So any infinitesimally small surface that can be (bijectively?) projected to a planar surface can be aligned in many copies to approximate a straight line ? Also, why does Smooth Infinitesimal Analysis article says small elements can't be curved? They can isn't it? I have reopened the post in Physics.SE, it has been migrated to math.SE though – niobium Dec 09 '24 at 15:04
  • It's not saying that a small line segment cannot be curved, it's saying something else that's more about the building blocks of its formulation. In calculus, infinitesimals aren't a well-defined concept on their own; instead it's based on limits. When differentiating, taking a small rise over run, what you're really doing on an intuitive level is you're "zooming in" as far as necessary, as if looking through a microscope, until the function looks like a straight line, in the limit. So you associate the slope of that straight line with that point. The differential itself can be thought of.. 1/2 – Filip Milovanović Dec 09 '24 at 18:03
  • ... as being a linear function in its own right, through the origin, having said slope, that when "bolted onto" the original function forms a tangent (and a linear approximation near that point). From what I understood, Smooth Infinitesimal Analysis kind of goes the other way around, packaging this information into these new mathematical objects that encode a position and a slope (described as tiny unbendable straight line segments, but that's just a visual metaphor, as they aren't really that, being infinitesimal in extent), and then builds functions out of those, instead out of points. 2/2 – Filip Milovanović Dec 09 '24 at 18:03
  • BTW, is your question something along these lines? As in, does it make sense to approximate an arbitrary surface and its boundary by a bunch of boxes, Minecraft-style (or using some approach similar to that)? – Filip Milovanović Dec 09 '24 at 18:41
  • @FilipMilovanović yes, my question is exactly about that. – niobium Dec 10 '24 at 09:05
  • But one answer says that in the $π=4$ or in the $\sqrt{2}=2$ cases, "we are approaching the area but not the path length". Should I draw the conclusion that the visual proof of Stokes' theorem is wrong in breaking down a macroscopic contour into many "yellowish" contours of very small sizes ? – niobium Dec 10 '24 at 09:34

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