At the current moment I'm reading a textbook, George F. Simmons' "Differential Equations with Applications and Historical Notes", and at first I had exactly the same question as was described here: Orthogonal Trajectories Using Polar Coordinates. Correct Calculations, Two Different Answers?
While that answer cleared some of my queries, this question bothers me;
In polar coordinates, why is the perpendicular is evaluated through: $\displaystyle{\frac{dr}{rd\theta}\to -\frac{rd\theta}{dr}}$ instead of $\displaystyle{\frac{dr}{d\theta}\to -\frac{d\theta}{dr}}$.
In my perspective, the differential equation of the family of original curves (in Cartesian coordinates) looks in the following manner:
If $\text{ }\displaystyle{\frac{dy}{dx} = f(x,y)}$, then the differential equation for the orthogonal trajectory of this family would
be, $\displaystyle{-\frac{dx}{dy} = f(x,y)}.$
Doing some research on my query, I found that answer. However, I did not understand the explanation.
I would really appreciate it if someone could explain the post in a more detailed way.