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In standard calculus, how can dx be an infinitesimal if infinitesimals do not exists in standard calculus

David G. Stork
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Serendipitous Epiphany
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    You could call $dx$ a differential, and then your question disappears $\ddot\smile$. – Angina Seng Sep 27 '17 at 16:13
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    The notation $\int f(x)dx$, the $dx$ is not explicitly an infinitesimal, it should be thought of as a reminder as to which variable we are integrating with respect to, not a manipulable object. – JMoravitz Sep 27 '17 at 16:13
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    I believe it's just notation. A lot of notation came before the rigor. – User203940 Sep 27 '17 at 16:14
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    In standard calculus, $dx$ is just a symbol (or two, for that matter). It can be interpreted as denoting an infinitesimal in suitable sense. – Wojowu Sep 27 '17 at 16:14
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    Infinitesimals are not rigorous in standard calculus, but that doesn't mean we ignore their existence. Non-standard is about creating a structure where math can be done to infinitesimals. – Kaynex Sep 27 '17 at 16:17
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    Recommend reading through the question and answers here. – JMoravitz Sep 27 '17 at 16:17
  • Since there aren't infinitesimals in standard analysis, $dx$ isn't an infinitesimal. Textbooks telling otherwise should be burned, and the ashes dissolved in acid. –  Sep 27 '17 at 18:13
  • @ProfessorVector: There are infinitesimals in standard analysis -- it's just that the real number system doesn't have any nonzero ones. The dual numbers, for example, are a standard construct. (and very useful, being closely related both to the tangent and the cotangent bundle on the real line) –  Sep 27 '17 at 18:22
  • @Hurkyl A standard construct, I see... non-standard analysis (not Robinson, but internal set theory, say) seems simple in comparison, though not without pitfalls. I hope you agree that both can't be taught to students of calculus, that would be cruel. You can't do that to mathematics. –  Sep 27 '17 at 18:47
  • @Lord Shark the Unknown then what exactly is a differential? Specifically, can it be rigorously defined in standard calculus. I know that dy = f'(x)dx, but then what is dx? – Serendipitous Epiphany Sep 27 '17 at 23:20

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