# When does infinitesimal notation break down?

## Main Question or Discussion Point

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 wont work in physics?

mfb
Mentor
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.

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.

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Wow great point. That really helps alot