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I'm studying calculus and sometimes I find it strange to treat dx(differential) like numbers! Substitution rule would be a good example. ( I will use the first example in this website http://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleIndefinite.aspx ) u = 6x^3 + 5 becomes du = 18*x^2*dx.

But I am curious whether there are mathematical grounds for treating differentials like numbers! If there are, can you tell me which field of mathematics deals with such kind of things?

Jin
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    Yes, though they were only developed in the 1950s, after Newton and Leibniz used them extensively. They are called Hyperreal numbers. See https://en.wikipedia.org/wiki/Hyperreal_number . Analysis that is based upon hyperreal numbers as opposed to limits is called Non Standard Analysis. – Jeevan Devaranjan Jul 04 '16 at 06:24
  • You want to read something about non-standard analysis. It uses different set theory than ZF, in the end it is more complicated than the standard analysis. – Masacroso Jul 04 '16 at 06:24
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    No, no, no, there's absolutely no need to bring nonstandard analysis into this. You can try looking up stuff about differential forms, or the word "nilpotent." – Qiaochu Yuan Jul 04 '16 at 07:01
  • @QiaochuYuan can you recommend me any websites or books? – Jin Jul 04 '16 at 07:03
  • Not really. The machinery used to make sense of this sort of thing is part of a bigger package (abstract algebra) that mathematicians learn somewhere between undergrad and grad school. I don't know a source that splits it off on its own. – Qiaochu Yuan Jul 04 '16 at 07:04
  • For integration by substitution, it all boils down to the chain rule really. $dy, dx, du$, for all purposes, can be regarded as no more than convenient notation. – MathematicsStudent1122 Jul 04 '16 at 07:24
  • Differentials are not numbers but functions defined on certain vector spaces and taking values, e.g., in ${\mathbb R}$. – Christian Blatter Jul 04 '16 at 07:44

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The grounds for treating infinitesimal $dx$ as a number is the rigorous theory of infinitesimals first developed by Abraham Robinson in the 1960s, based on earlier work by Thoralf Skolem, Edwin Hewitt, and Jerzy Łoś.

The OP asked for sources where one can learn about this approach. The best source is the textbook Elementary Calculus by Keisler where this is explained at a level accessible to a freshman.

Leibniz used infinitesimal numbers extensively specifically in the form of differentials which he denoted $dx$, $dy$. As it turned out his procedures and his theoretical justification for manipulating these were more soundly based than Bishop Berkeley's criticism thereof; see this site for abundant literature on the subject. In my teaching experience this approach is more successful with the students.

Mikhail Katz
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