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can argue that it is the ratio "up to infinitesimal error", but justifications in that weaker sense were well known and widely taught before nonstandard analysis (e.g., as "up to an error tending to 0", up to o(1) error, up to higher-order terms in a series, up to a 1+o(1) factor, equal to leading asymptotic order, ...). The formalism where dy/dx is, exactly and not up to small error, the derivative, is
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they probably had a different way to write multiple fractions. Anyway, I would be happy if you would incorporate the notation you propose as an addition (but not a replacement) to the text, since it certainly makes sense, and is certainly now used. It could comfortably be inserted into the explanation for the origin of the d^ny/dx^n notation.
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I removed the statement that nonstandard analysis gives a new ratio interpretation or justification of dy/dx. What it provides are infinitesimal numbers, and thus infinitesimal increments (differentials) of functions, but the derivative does not become more of a ratio than it was using limits. You
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But if it is a functor, the argument should be anonymous, which does not match the d/dx notation very well. However, f(x) in df(x)/dx is not constant, because x is not constant. (If x is constant, you are dividing by zero, which is not allowed. You have to fill in the value of "x" after computing the
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I suspect that the notation you suggest was not current in the 17th century, since it presumes the (later) idea of d/dx as an operator, and it was probably for precisely this reason that a more concise form was needed for multiple derivatives. However, I'm not an expert of 17th century notation, and
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On second thought, it can be argued that for a page dealing with notation, and not formal mathematics, a simplified, less precise form is acceptable as long as it is clear and unambiguous to someone with a semester of calculus (or limits). I guess the notation for integral is then acceptable, but I
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I really don't get why the derivative is supposedly not a fraction since it is said to be the limit of a fraction. Non standard analysis is interesting but it seems it allows to precise the meaning of the derivative, not contradicting its "nature". Several times in the article, it's repeated a
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I have recently been cleaning up this page and adding new material and references. As I look at the structure of the page, however, I am a bit concerned by the amount of material on non-standard analysis that appears in the lead. I am considering the following changes:
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The modern formalism for derivative and integral is imprecise. Unless there is a reason not to do so, we should use that given in any real analysis textbook, namely the
Riemann sum for an integral and the long functional forms of y(x) and x for the derivative.
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derivative is not a fraction without any counter example, and then it's explained it's the limit of a fraction.I have nothing against subtlety, but I would be grateful if the subtleties were clearly exposed, and not let "as a trivial exercice for the reader"
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I think that these changes might restore some balance to this article. Whether or not I do this, I do plan on expanding the history section to include some of the alternative notations that
Leibniz played around with (based on Cajori's treatment). Comments?
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This article needs a lot of work. The second paragraph (not the line, the paragraph) is nearly incoherent. The closing statement describing units merely hints at what I wanted to know. Maybe it's somewhere else; in that case, a link will be needed.
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Notation for integration is mentioned at the very end of the article, but it wasn't introduced anywhere. We should add details with examples, as
Leibniz notation is also used to describe integration along a path.
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Pull all the rest of the non-standard analysis out of the lead and the history section and create a new section with that material and place it before the last section on other notations.
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445:{\displaystyle {\frac {\mathrm {d} {\Bigl (}{\frac {\mathrm {d} \left({\frac {\mathrm {d} \left(f(x)\right)}{\mathrm {d} x}}\right)}{\mathrm {d} x}}{\Bigr )}}{\mathrm {d} x}}}
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990:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\Bigl (}{\frac {\mathrm {d} }{\mathrm {d} x}}{\Bigl (}{\frac {\mathrm {d} f(x)}{\mathrm {d} x}}{\Bigr )}{\Bigr )}}
573:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}{\Bigl (}{\frac {\mathrm {d} }{\mathrm {d} x}}{\Bigl (}{\frac {\mathrm {d} f(x)}{\mathrm {d} x}}{\Bigr )}{\Bigr )}}
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Google Books "Mathematischer, naturwissenschaftlicher und technischer
Briefwechsel von Gottfried Wilhelm Leibniz" Letter No. 44, Leibniz to Bernoulli, p.121, l.17
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Having gotten no comments in a week and a half, I will proceed to make these changes and I am sure that if I ruffle any feathers I will hear about it. --
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into this article. A while ago, I also adapted the notation here for coherence with that article. There is now a new (not fully developed) article at
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Put into the lead the simple statement that non-standard analysis can be used to legitimize
Leibniz's original conception of differentials.
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For those like me wondering whether the d should be upright or italic, this has been discussed at length on
Knowledge before: see
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articles, where the issues are more straightforward, because Newton's notations for integration are not developed in wikipedia.
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as functor applied to functions rather than to function values. Function values are considered as constants and so
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developed rigorous mathematical explanations for
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The year is wrong, it must be fixed but I'm not allowed to do so, can someone else look into the year issue?
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Did
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Knowledge. If you would like to participate, please visit the project page, where you can join
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In the absence of any talk here, I have implemented the proposed merger by moving the content of
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Added also that Leibniz notation is used in integrals, which is where it is rather important.
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Please help fix the broken anchors. You can remove this template after fixing the problems. |
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based on these ideas. Robinson's methods are used by only a minority of mathematicians.
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And here is also my problem with this notation. The modern version regards
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derivative, similar as the same way you do with limits calculations.) --
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Good job. I have a question about this odd looking thing:
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1321:82.36.30.34
1272:—Preceding
262:Fresheneesz
212:—Preceding
112:Mathematics
103:mathematics
59:Mathematics
1692:Categories
1574:Regarding
246:Needs work
1392:Jerzy Łoś
1329:What did
274:GangofOne
254:-Malakai
238:SamuelRiv
218:SamuelRiv
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1406:wrote a
1274:unsigned
226:contribs
214:unsigned
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190:before.
139:on the
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1364:Tkuvho
296:Merger
282:GIYF:
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1561:Vyvek
1350:zzo38
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287:TomJF
1680:talk
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222:talk
171:Tip:
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