Fractions of differential forms

In the classes of an undergrad mechanics course, we were shown a cheeky trick to solve differential equations. If we had an equation of the form

we were told to treat the derivative as if it was actually a fraction. That allowed us to multiply by and to obtain the equation

This looked a bit odd to me then; what do the and even mean in isolation? Depending on the tutor, those were “very small steps” or “purely formal symbols”. It didn’t matter much, since the next step was to integrate both sides, often leading to the correct solution. So I filed this as a somewhat odd but convenient computational trick.

I learned about differential forms much later in my mathematical career. This is generally regrettable: I think they should be moved to the very start of every applied math curriculum. But at that point, there was a candidate for giving the isolated and a definite meaning.

The first step is to unpack some applied math conventions: We can instantiate the variables and into mathematics by conjuring up some smooth ambient space and letting , be smooth functions . We can differentiate by because is a variable of , which concretely means that there is some smooth function such that .

Now and are concretely differential -forms in . Via the chain rule we can calculate that:

If these were mere numbers, we could divide this equation by . We can’t divide by a differential form, but we can use the equation above to motivate a definition of the differential quotient:

We then get the pleasing equation:

With this notation in place, the separation of variables trick from undergrad now becomes a well-defined statement about differential forms:

No very small steps or formal symbols necessary.

We can also show the inverse of this implication! Suppose that we have smooth functions and smooth functions so that . Consider any point so that and . Then the map is a submersion and by the submersion theorem there exists a coordinate system for an open neighbourhood of so that . We may shrink so that on all of . Then

The differential forms form a basis of and so

for all . Therefore is a function of and so the differential quotient is well-defined by our construction above, satisfying

Because is a basis element of , we have

We conclude that at any point where the right hand side is well-defined, meaning and , we get the implication