Time is the essential variable in the science of mechanics-velocity is dx/dt, acceleration is dv/dt. Suppose we were to invent a different derivative-hold time constant and measuer dx; so that now we have dt/dx;dt/dv. Would mechanics be the same?
I’m not sure what you’re getting at here. dt/dx is simply 1/v. If you plot x vs t, dx/dt is the slope, while dt/dx is the slope if you plot t vs x. Neither of them are being held constant, the differentials mean that both x and t are changing an infinitesimal amount.
What? Jesus. What? “Different derivative-hold time constant”?
You can express any of the derivatives of any of the functions with respect to any of the variables in mechanics and it all works. I don’t think the dx/dt and dt/dx notation is very helpful in this context, because it looks like division, and worse yet it is in some ways related to division, until you take d2(x)/dt^2. If you treat dx/dt or dt/dx as fractions you might produce something nutty, I guess. But, mechanics is perfect, and there isn’t anything so broken so early on in the calculus that changing which things you consider depending on which other things will make the calculus go wrong.
Yes, I do. First Click, then Clack.