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I think the title pretty much explains what I want to ask. Bassically is $$\lim_{x\to 0}(x)$$ different from $dx$?

Anoher way to put this would be, how wouldth equation: $$F=adm$$ Be different from $$F=a\lim_{m\to 0}(m)$$ I think the second can be written as $$F=0$$ Whilst the first we cannot, but I cannot explain why?

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    $\begingroup$ Whatever else $dx$ is, it is not a real number. The limit of $x$ as $x \to 0$ is a real number. These two quantities cannot be equal. $\endgroup$
    – Simon S
    Feb 20, 2015 at 19:28
  • $\begingroup$ @SimonS sorry why is $dx$ not a real number? Or equivlently If I have an infitismal bit of mass $dm$ why is this not a real number? $\endgroup$ Feb 20, 2015 at 19:29
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    $\begingroup$ I was hoping that the title didn't explain well what you wanted to ask. $\endgroup$
    – user207710
    Feb 20, 2015 at 19:29
  • $\begingroup$ Joseph, how do you define an infinitesimal? How do you define arithmetic operations on those? Now, do these definitions fit with the usual real numbers definitions? This should answer your question. $\endgroup$
    – bartgol
    Feb 20, 2015 at 19:32
  • $\begingroup$ @bartgol I get your (and Simon S's) point but I can't see when we are using it to define real quantites, such as an infitimsal bit of mass, or an infitismal length of string, how it cannot be a real number? $\endgroup$ Feb 20, 2015 at 19:34

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It is confusing because the way derivatives are taught today are different from how it was done back in the 1600s. Back then a derivative was $dy/dx$, where $dy$ and $dx$ were infinitesimal quantities. Today, we introduce derivatives using limits. The confusing part is that learn differential calculus by limits but use the old notation.

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