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In a few of my courses in mechanics certain statements/equations have been proved by assuming that two infinitesimals multiplied by each other are zero.

For instance in the equation : $dx + dy + dx^2 + dx.dy$ , both $dx^2$ and $dx.dy$ are assumed zero.

I understand that in an order of magnitude sense these are significantly smaller and can hence be ignored but are these quantities actually 0 due to being infinitesimal? If so, how do we go about proving it (or is it obvious) and if not are all the equations which use this assumption simply approximations to reality and induce slight error?

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A differential like $dx$ in an equation like $$dz = dx + dy + dx^2 + dx\,dy$$ is shorthand for a quantity that will eventually go to zero in a limit. If a differential is divided by another differential, then it has a chance of resulting in a finite, non-zero quantity. Let's assume $x$, $y$, and $z$ can be parameterized by a variable $t$. If we divide the above equation by $dt$, $$\frac{dz}{dt} = \frac{dx}{dt} + \frac{dy}{dt} + \frac{dx}{dt}dx + \frac{dx}{dt}\,dy.$$ In the limit where $dt$ goes to zero, $dx$, $dy$, and $dz$ go to zero as well. This means that the last two terms are finite quantities ($\frac{dx}{dt}$) multiplied by zero ($dx$). Hence, we cross them out.

Mathematicians hate us physicists for doing things like this.

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