Originally Posted by

**DrRocket**
They are right.

The reason that they tell you that dx/dt=1 should not be converted to dx=dt is because in calculus classes the expression dx has no meaning. dx/dt is not a fraction.

There are ways to do this rigorously, but in calculus classes you are not prepared to discuss them. And even if at some point you get to the point where it starts to make sense, derivatives are still not a fraction.

Physicists do mathematical manipulations that often do not make sense. Sometimes they get away with it, largely because there are means that they don't understand whereby that manipulation can be justified in a larger context. But sometimes they just get nonsense. And often they don't know which without a physical experiment.

In the case of dx/dt =1. You know that then x =t + constant. So you formally use

dx or dt interchangeable in an integral, as it is just the change of variables process that you learned in calculus.

So you get to something that is correct by a process that is on the surface quite nonsensical. Basically compensating errors.

But also as I said there is a way to do this with more rigor, but to do that you have to have a real definition for what dx means by itself. For that you will have to wait until you have quite a bit more mathematics under your belt.

You might at some pont want to pick up Michael Spivak's litter paperback book *Calculus on Manifolds* where you can see the use of "dx" much more rigorously. You can read that book once you have pretty well mastered calculus and are ready for a modern treatment of calculus of several variables, or what is commonly called "advanced calculus". A good time to do that is right after you have taken a first course in electromagnetics and have seen some vector analysis (div,grad, curl etc and what are called the Divergence Theorem and Stokes Theorem in those classes). Spivak will show you that the Divergence Theorem, Stokes Theorem and the Fundamental Theorem of Calculus are really the same thing and can be generalized to what mathematicians all call just Stokes Theorem and that it applies in all dimensions.