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| | ::: No it isn't "basically" what you said. I know FTC applies to a wider range of functions, that's why I said "restricted to smooth functions". I have not even the slightest idea what process you're trying to explain or why you're talking about 2D integrals. FTC restricted to smooth functions ''is exactly'' Stokes restricted to a line-segment, there is no "process". Again, [https://en.wikipedia.org/wiki/Stokes%27_theorem#Introduction this wikipedia section] explains this quite well, albeit informally. [[User:Zmatt|Zmatt]] ([[User talk:Zmatt|talk]]) 11:01, 10 September 2018 (UTC) | | ::: No it isn't "basically" what you said. I know FTC applies to a wider range of functions, that's why I said "restricted to smooth functions". I have not even the slightest idea what process you're trying to explain or why you're talking about 2D integrals. FTC restricted to smooth functions ''is exactly'' Stokes restricted to a line-segment, there is no "process". Again, [https://en.wikipedia.org/wiki/Stokes%27_theorem#Introduction this wikipedia section] explains this quite well, albeit informally. [[User:Zmatt|Zmatt]] ([[User talk:Zmatt|talk]]) 11:01, 10 September 2018 (UTC) |
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| − | :::: You seem to be missing what I am referring to. There are at least two parts. Let's start with the main one. You keep referring to the same place of the same article. That is not under contention, so it is irrelevant. So I checked Wikipedia's article on FTC itself and I think I see why you don't see my point. When I learnt FTC from textbooks, the definition of integrals is via the area under curve, i.e. the relevant bit in the FTC article is the geometric intuition. The FTC article, however, quite much like you seem to be, however, only covers the anti-derivative part. In a sense, it comes down to the definition of what an ``integral" means. AFAIK, for beginners, there is only 3 definitions in common use, the directed area under curve, limit of a certain sum, and anti-derivatives. When I teach, I tend to define the directed area under curve, just because students like to see things. Because of that, my FTC has to cover the area under curve. And that is the 2D integral known to Leibniz. If you want the 2D integral, then you ought to integrate the fundamental differential form I was talking about---you don't talk about generalised Stokes's theorem without differential geometry, and I am trying to say that the identification of a definite integral with the area under curve is what is taught to beginning students, but is highly non-trivial under differential geometry! I hope this is clearer. Of course, the moment the 2D integral is reduced to a 1D integral with new boundaries, then the part you keep referring to is relevant, and again, not under any contention. I am simply saying I am not happy with that being the sole content of FTC. The FTC I respect is the one that includes the geometrical intuition. Finally, just the quibble---what part of my ``the restricted FTC is a consequence of generalised Stokes's theorem" is different from your ``FTC restricted to smooth functions is simply a special case of Stokes' theorem"? Even if you disagreed with my ``applied twice", you should not be disagreeing with my ``basically what I said". [[Special:Contributions/162.158.166.191|162.158.166.191]] 16:55, 10 September 2018 (UTC)
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| | "Munroe's theorem" should definitely refer to the circle thing in the alt text {{unsigned ip|162.158.62.57}} | | "Munroe's theorem" should definitely refer to the circle thing in the alt text {{unsigned ip|162.158.62.57}} |
