Definition talk:Differential of Mapping

The previous version of the page utilized only concepts in a Calculus I curriculum. This new one is no longer accessible to students new to Calculus (and I don't know what a Frechet derivative is). --GFauxPas (talk) 18:53, 28 September 2012 (UTC)


 * Tried using Google? --prime mover (talk) 18:55, 28 September 2012 (UTC)


 * I did, and it utilizes concepts that are too advanced for me. The main point was the first part of my comment, that this page is no longer accessible to people new to Calculus. --GFauxPas (talk) 18:59, 28 September 2012 (UTC)


 * If by that you mean "by people who don't have the particular textbook that you had access to in college", I'm afraid it probably never was ... --prime mover (talk) 19:03, 28 September 2012 (UTC)


 * I don't think it's particular to my textbooks. The new definition utilizes open sets and bounded linear operators. --GFauxPas (talk) 19:10, 28 September 2012 (UTC)


 * Beg your pardon. I see what you mean now. It has indeed become seriously inaccessible. --prime mover (talk) 19:19, 28 September 2012 (UTC)


 * Agreed the Frechet derivative is post-undergraduate. It's the same thing as all the usual stuff in finite dimensions. I used it because Definition:Derivative only has one dimension, it wasn't really a good solution. --Linus44 (talk) 20:32, 28 September 2012 (UTC)


 * I'm an undergraduate. This page reminds me of Definition:Polynomial. I didn't know what it was talking about at first. But everything was linked and what was once cloudy very soon became clear.


 * I also want to say that on several pages (Definition:Integer and Definition:Sequence for example) you have an informal subsection for accessibility. If you can use this new definition here to prove results it should be preserved somehow. Just my 2 cents. --Jshflynn (talk) 20:50, 28 September 2012 (UTC)


 * It's hard to know where the line is between demanding the reader expects some opacity in the definitions, and just being more confusing than is necessary. Def::Polynomials is a good example; the definition is completely obscure, and only exists so we can say "all of that stuff we assumed we can do is fine", and from then on you can stop worrying.   As you say, an informal introduction before the more pedantic stuff is a good solution

In fact, there is a more general problem: at the undergraduate calculus (for me) was all done from $\R^n \to \R^m$. That is, it was done for vector spaces, but only identifying them all with $\R^n$. Generally it'd be nicer to have the results formulated without fixing a basis like this, and the translation of results doesn't mean anything more than fixing an identification with $\R^n$, but it would probably be unhelpful when trying to learn the stuff.

So should everything be done in just $\R^n$? Or just vector spaces? It'd get pretty messy doing it for both, and having a load of pages saying after fixing a basis the results are the same.

My opinion: calculus should all be done in $\R^n$. Then anything that might be put under Category:Differential Geometry should be done in vector spaces, rather than repeating everything, and having a load of somewhat-trivial-but-still-kind-of-necessary-for-rigor linking pages. --Linus44 (talk) 21:55, 28 September 2012 (UTC)