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Even if many mathematicians don't like the notation, I have found in many rigorous math books things like




What I don't understand is this: under which conditions is ok to treat infinitesimal as numbers ? (multiplying for example both sides of an equation by dx).


Not only, what allows me to integrate both sides of the last equation ???


marked as duplicate by Mark S., Hans Lundmark, Kemono Chen, Gibbs, Dylan Feb 11 at 14:34

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If you want to make this rigorous you can introduce a number system similar to that of dual numbers, viz. $dx^2=0$. (See also here.) Depending on the calculus task at hand in more advanced examples, you might change these axioms slightly. For example, a metric $ds^2=g_{\mu\nu}dx^\mu dx^\nu$ would use $dx^\mu dx^\nu dx^\rho=0$, while Brownian noise in stochastic calculus may be taken to satisfy $dW_t^2=dt,\,dW_t^3=0$.


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