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I know $\frac{dy}{dx}$ may be a shortcut of $\lim_{h \to 0} \frac{y(x+h)-y(x)}{h}$, which is totally rigourous, but it loses that sense if I write $dx=f\cdot dy$.

How could that become rigourous again and in what frame?

Jonas Daverio
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  • Have a look at differential forms. – md2perpe Feb 16 '19 at 15:33
  • There are other ways of rigorously defining differential forms. I recall one discovered by Solomon Leader based on the Henstock-Kurzweil integral that allowed you to do pretty much everything classical mathematicians did with differentials. For example $ds = \sqrt{dx^2 + dy^2}$ is a perfectly sound definition of $ds$. (This expression doesn't make sense using the differential forms from md2perpe's link.) – Paul Sinclair Feb 16 '19 at 23:00
  • Why do you say it doesn't make sense? Can't I define $ds(\vec{v})=\sqrt{(dx(\vec{v}))^2+(dy(\vec{v}))^2}$? – Jonas Daverio Mar 02 '19 at 18:27

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