What Do Mathematicians Think of Physicists Cancelling dx?

[ShayanJ]

In some places in physics like proving the wave equation,we see that the physicist divides the equation by dx or \partial x or maybe cancels some of them and then says with a smile:You shouldn't tell a mathematician!
I want to know the ideas of mathematicians about that and want to know why its wrong?
thanks

 

[HallsofIvy]

"dy/dx" is NOT a fraction as defined so "dy" and "dx" are not separate values. However, because it is defined as a limit of fractions, we can always "go back before the limit, use the fraction property, then take the limit again". So "dy/dx" can be treated like a fraction and to make that precise, mathematics defines "differentials", dy and dx, which, while not really the derivative have the property that the derivative, dy/dx, is the fraction "dy" divided by "dx".

It is perfectly legitimate, mathematically, to "divide by dx" provided you have checked all of the properties required to be sure the differentials exist. It is that part that physicists typically play "fast and loose" with, assuming that, since everything has a specific physical meaning, they must exist.

 

[Skrew]

The only thing you need to watch out for is dividing by zero(such is the only minor problem with the chain rule proof using dz/dx = dz/dy*dy/dx

 

[ShayanJ]

The only thing you need to watch out for is dividing by zero(such is the only minor problem with the chain rule proof using dz/dx = dz/dy*dy/dx

I don't understand what could be that zero!