When does infinitesimal notation break down?

[jaydnul]

Everything I've encountered in physics so far allows infinitesimal numbers to be manipulated as real numbers. But there has been much criticism towards Leibniz's notation, and I assume it is for a reason. When in mathematics will the infinitesimal notation not work? Including treating $\frac{dy}{dx}$ as a fraction to solve differential equations and such. Does it ever breakdown? If so, are those purely mathematical, scholarly problems, or is there places where the notation won't work in physics?

 

[mfb]

It breaks down if you have curves with a tangent parallel to the y-axis, for example.
And there are functions of multiple variables where the order of differentiation matters.

I don't think I ever had that problem with physics-related equations.

 

[brmath]

When you use dy/dx as a fraction in separating variables, you are not really doing fraction arithmetic. You are using the chain rule, and the separation of dy and dx is a shortcut way of writing that. To see that, let's consider the example dy/dx = y. Using separation of variables you would write dy/y= dx. Then you integrate both sides, the left with respect to y and the right with respect to x (also a rather suspicious maneuver) and you get log y = x + c or $$y = Ae^x$$ where A is some constant. This is a handy way to get the right answer, but here is what you are really doing:

Let y = f(x). Your equation dy/dx = y can be rewritten in this notation as f'(x) = f(x). This gives us f'(x)/f(x) = 1. Integrating boths sides with respect to x (a clearly legitimate maneuver) we get

log f(x) = x + c
log y = x + c and so
$$y = Ae^x$$

The chain rule comes in when you observe that $$\frac{d}{dx}logf(x) = f'(x)/f(x))$$.

So the general answer about when you can split dy and dx and treat dy/dx as a fraction is that it is justified when it is a shortcut way to use the chain rule.

 

[jaydnul]

Wow great point. That really helps a lot