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I'm a CS grad who is fairly strong with maths but never really did learn to write a proper proof. All of these questions I would have been able to "solve", but
by 0xfaded 10y ago
I'm a CS grad who is fairly strong with maths but never really did learn to write a proper proof. All of these questions I would have been able to "solve", but the proofs would have been very casual. What would be a valid answer it 2.3 where it basically asks you to integrate an expression of F after substituting y'=dx/dy?
- Smaug123 10y ago"2.3" is a bit ambiguous as a label; do you mean part 3 of question 2, or the part of question 2 in which equation 2.3 occurs? If you mean part 2 of question 2: my answer would be "dF/dx = del F/del y dy/dx + del F/del y' dy'/dx; but we know del F/del y = d/dx(del F/del y'), so dF/dx = d/dx(del F/del y') dy/dx + del F/del y' dy'/dx, which by the product rule is d/dx(y' del F/del y'). Now integrate both sides." If you mean part 3 of question 2: just substitute F(y, y') from (2.2) into (2.4).
- mwane 10y agoWhat is the difference between d and del? > but we know del F/del y = d/dx(del F/del y') ????
- Smaug123 10y agodel refers to the partial derivative, with LaTeX symbol \partial. d refers to the total derivative. "del F/del y = d/dx(del F/del y')" is just the Euler-Lagrange equation.
- mwane 10y agoWhich 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.
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- 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 $\dfrac{d}{dx}$.
- mwane 10y agoI think a major problem with mathematical language is lack of discoverability. If I don't understand a specific symbol or notation, there is no way to click "go to definition" or look it up on Google. Furthermore while mathematicians pride themselves on rigor, I would bet that a lot of proofs require the reader to fill in incomplete steps and/or contain notational mistakes. Just try to imagine what kind of code you would end up with if you did not have access to a compiler and relied only on your logical reasoning for writing correct code. Even if the code was reviewed by other peers, who themselves do not have access to a compiler, they would miss some mistakes because, being familiar with the subject, their brain would fill in the correct meanings.