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So I have a system $\frac{dx}{dt}=f(x,y); \; \frac{dy}{dt} = g(x,y) \\$. I am wondering if I can calculate the slope of the quivers for my phase portrait as $\frac{dy}{dx} = \frac{g(x,y)}{f(x,y)}$ ? It makes perfectly good sense to me to do it this way, but I just read a pretty good write up here on Math.SE and saw the Leibniz notation described as a limit. Here is the link: Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? This article made me feel as though this may not be safe to do, because I would essentially be cancelling out "dt"s

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Julian Cienfuegos
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The slope $\frac{dy}{dx}$ is in fact not a ratio (it is the limit of a ratio), but it is true that $\frac{dy}{dx} = \frac{g(x,y)}{f(x,y)}$ whenever $f(x,y) \neq 0$, for in that case

  • from the inverse function theorem $x(t)$ has a local inverse whose derivative is $\frac{dt}{dx}=\frac{1}{\frac{dx}{dt}}=\frac{1}{f(x,y)}$;
  • from the chain rule $\frac{dy}{dx}=\frac{dy}{dt} \frac{dt}{dx}=\frac{g(x,y)}{f(x,y)}$.
P. Gomes
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