Here, by passive transformation, I mean the transformation of the coordinate system.
While watching the following video on YouTube on coordinate transformation,
(https://www.youtube.com/watch?v=XtpVVcKXfnA&list=PLdgVBOaXkb9D6zw47gsrtE5XqLeRPh27),
the author mentioned three conditions for a set of transformation equations to satisfy in order to be a coordinate transformation. 1. All functions must be real valued. 2. All functions must have continuous second partial derivatives. 3. All functions must be invertible.
I do intuitively understand the conditions 1 and 3. But regarding the second condition, I don't understand the reason.
To summarize my question, If, $\overline{x_i}=f_i(x_1,x_2,x_3,...x_n)$ where the integral index $i$ varies from $1$ to $n$, be a set of $n$ transformation equations from $\mathbb{R}^n$ space to $\mathbb{R}^n$ space, with $f_i$ being bijective functions, how do we determine if the transformation is a coordinate transformation and not an active transformation (one that changes the shape of an object defined in a specific coordinate system)?
Well, the reference video says that the functions $f_i$ must have continuous second partial derivatives. That must count in all the second partial derivatives (mixed and unmixed), in which case, Clairaut's theorem will hold. Thus, at any point in the coordinate system, $f_{{x_i}{x_j}}=f_{{x_j}{x_i}}$. How does that determine if the transformation is a passive and not an active one?