5 ms·
Which means there is no real difference; if all the types are known, it's just some form of Hungarian notation?
by mwane 10y ago
Which means there is no real difference; if all the types are known, it's just some form of Hungarian notation?
- piefoot 10y agoif a function depends only on multiple variables then those are partial derivatives, if the function depends only on one variable then there is only one derivative d/dx
- mwane 10y agoI get that, but ultimately, why does the notation distinguish between the two classes "function with zero arguments held constant" and "function with n >= 1 arguments held constant"? It seems somewhat of an arbitrary distinction, when in fact the value of n is always clear from the context.
- piefoot 10y agoI think that the difference is from historical reasons, anyway when if you introduce in school partial derivatives with functions of only one variable that is confusing because here partial is not partial but global.
- sidusknight 10y agoI think his point is that you can always use the partial derivative notation and be correct.
- deleted 10y ago[deleted]
- deleted 10y ago[deleted]
- Smaug123 10y agoNo, you can't do this. The Wikipedia page [1] contains a simple example which hopefully makes things clearer. [1]: https://en.wikipedia.org/wiki/Total_derivative https://en.wikipedia.org/wiki/Total_derivative
- mwane 10y agoWell, is f really a function of x and y when only one of them can be chosen? Seems like an abuse of notation that introduces unnecessary special cases. Abstractions (here: a function) should be simple and well-defined, not overloaded by natural language.
- Smaug123 10y agoStrictly, if one knows in advance that y is a function of x, then one should write f(x, y(x)) to make the dependency clear. This just gets unwieldy, though, as anyone who's ever programmed in Mathematica can tell you. However, what if a priori we only know that f is a function of two variables, and then we decide to evaluate it along a path {(x, y(x)): x in reals}? Under your scheme, we'd have to define a new function g(x) = f(x, y(x)) and reason only about g, which is even more painful, not least because maybe g(x) only depends on x through y and could be written more succinctly as g(y(x)). The current notation is simply more flexible and easier to use, and it remains unambiguous because we have access to these two operators $\dfrac{\partial}{\partial x}$ and $\dfrac{d}{dx}$.