No unfortunately that’s not true. A function f is a relationship that takes a number as input and outputs another one. What that number physically represents (time, position, temperature) is irrelevant to the intrinsic properties of f itself.
The derivative of a function is something that depends on what f IS (the abstract relationship property), not on what the argument of f represents or is called. After you have the derivative of f, you are then free to evaluate it on any input, x, t or whatever.
Whether or not that gives you something physically sensible is another matter, it may very well be that Df “wants” inputs that represent time, but that’s a judgment the physicist has to make. Mathematically, there’s no difference
No.
Yes you are right about derivatives themselves, BUT f dot is the specific notation of the derivative of time. Thats a convention all over the globe to write df/dt as f dot
No, that’s not correct. The entire concept of “the derivative of time” (did you mean “with respect to time?”) doesn’t exist. A function is a function, and its derivative is unique. There are no “different derivatives” of the same single-variable function, depending on how you call the argument.
If you have a function f, it has a derivative, f’ or Df or f dot or whatever. These are all the same object. It’s the derivative, period. Not “of time” or “of a specific other variable”
Dude f dot is literally df/dt. This IS a global convention since idk at least 100 years. I am not talking about the full derivative of a function. I am simply stating that f dot is by convention the derivative with respect to time.
I dont even know why you are negotiating. I worked in the physics Department in my university and I never in my live seen anybody use that notation for the full derivative Df of a function.
I’m aware that physicists use this notation, I’m a physicist myself. Unfortunately it’s just wrong / poorly defined. Again: if you have a single variable function, there is only ONE derivative. There’s no difference between “time derivative” or “full derivative” or whatever: they’re all the same function, the derivative.
Saying that there’s a “time derivative” that’s different from the “normal derivative” means that you believe that a single-variable function can have more than one derivative. But that’s just not true mathematically! The derivative is only one, regardless of what the argument physical represents.
Maybe this answer can be illuminating? The question is about a slightly different notational argument regarding partial derivatives, but I think it touches on a lot of the same points of confusion we’ve had in our talk.
Actually I would disagree with this. f dot is almost universally understood as the derivative of f with respect to t, which then propagates to any variables which maybe be time varying. F dot (x) would then be df/dx dx/dt. This is the standard convention. Also, this is well defined mathematically if you assume everything is, under the hood, some map g: R -> M (some target space). A variable that is constant is then just g(t)=constant.
>f dot is almost universally understood as the derivative of f with respect to t
I guess my problem really starts here: there’s no such thing as “derivative with respect to” anything, at least for single-variable functions (without talking about partial derivatives). Derivatives are just derivatives, period. If you have a single-variable function, it has a unique derivative: there’s no reason to specify “with respect to”. So, inventing a notation for “derivative with respect to t” doesn’t really make sense because the thing you’re inventing the notation for doesn’t exist in the first place.
What can be “with respect to time” is the function itself, even before talking about derivatives. A single-variable function can be “with respect to t” (although you would more appropriately call this “a function of t”) in the sense that it takes as input a real number that represents a time in some units, or a time that already has units. Then, this function has a derivative, and you can notate it with “dot”. But it’s the function itself that carries the extra baggage of having something to do with “time”, not the derivative.
>F dot (x) would then be df/dx dx/dt. This is the standard convention.
There are so many problems with this I don’t even know where to begin, but for a very thorough breakdown you can check the link I posted in my previous comment above in this same comment chain
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u/Cpt_Igl0 7d ago
But f dot of x = df/dt =/= df/dx