[eaglefield]

The solution of differential equations by separation of variables in physics is also notated in an abusive way. You have some differential equation

dy/dx = g(x)h(y)

You separate the variables by some quick manipulations

dy/h(y) = g(x) dx

And then you have a small step in some coordinate on both sides. So by integrating both sides

\int 1/h(y) dy = \int g(x) dx

you find a solution to your differential equation. Obviously there's a real formal procedure underneath it with also some safeguards. For example you're supposed to check that h(y) doesn't equal 0 at any point. But the happy path in physics is often done without worrying about all that.

[d4rkn0d3z]

The real formal procedure:

dy/dx = g(x)f(y)

Let h(y) = 1/f(y)

=> dy/dx = g(x)/h(y)

=> h(y) dy/dx = g(x)

Now, we integrate both sides,

int h(y) dy/dx dx = int g(x) dx

But the left hand side is the same as

int h(y) dy by substitution rule of integration.

Therefore,

int h(y) dy = int g(x) dx

Proceed with solving now, no abuse since the substitution rule is provable. QED

[tptacek]

Yes! Separation of variables the other instance in the back of my mind. I suck at math (I've had basic ODEs for just a couple months now) but are there more examples like this?

I find this whole topic very gratifying because Leibniz notation seems very arbitrary and I'm glad it's not just me. :)

[leephillips]

More examples? Any undergraduate text in thermodynamics. The entire way the subject is taught depends on treating differentials as numbers. Even in partial derivatives.