Talk:Chain Rule for Real-Valued Functions

I know I suggested "real-valued function" for a function from $\R^n$ to $\R$ but such a definition does not only encompass such functions. A "real-valued function" (go check it) is a function from any set to the real number line. My suggestion was for describing such a function in the body of the text for providing a link. In the context of a page title I think something more specific is called for. --prime mover 01:09, 3 April 2012 (EDT)
 * Maybe 'Euclidean mapping' or 'Euclidean function'? --Lord_Farin 02:54, 3 April 2012 (EDT)
 * Specifically it's from a vector - would that cover it? --prime mover 07:49, 3 April 2012 (EDT)
 * I would suggest Larson's "Real-Valued Function of Several Variables", except you don't like the connotation of a vector being several things. So I won't. --GFauxPas 08:36, 3 April 2012 (EDT)
 * Sorry, but can you cite where I gave that disapproval?
 * I think what I'm not keen on is putting this result up onto the domain of vectors specifically. The partial differentiation definition page just took an ordered tuple of arguments. Clearly, having established the result for the ordered tuple, you can then apply it to when the ordered tuple defines a vactor - but to specifically apply this result to a vector to start with might be limiting. And as I've said before, I have reservations about Larson. --prime mover 09:22, 3 April 2012 (EDT)
 * "In particular "function of several variables" is more a description than a definition. It's clear from the notation that the domain is $\R^n$. Anyway, thinking of the elements of $\R^n$ as separate entities makes you lose track of the fact that an ordered tuple or vector is an entity in its own right. --prime mover 01:19, 2 April 2012 (EDT)"
 * I don't have any inclination for or against Larson - that's just the book I have. You can do whatever you need to do to this page for you to feel content about it. --GFauxPas 09:30, 3 April 2012 (EDT)
 * I can't remember what the context was in which I said that. Can you give me the link to the page on where that came from? (That's what "cite" means.) --prime mover 10:14, 3 April 2012 (EDT)
 * Oh my b, sorry. http://www.proofwiki.org/w/index.php?title=User:GFauxPas/Sandbox&direction=prev&oldid=87623 --GFauxPas 10:34, 3 April 2012 (EDT)

The point is:
 * a) The actual term "function of several variables" is more vague (general, woolly, whatever) than the concept of a function of an element of $\R^n$, which is why I was not so keen on using those specific words.
 * b An element of $\R^n$ is not a vector unless it is specified that $\R^n$ is the vector space represented by $\R^n$ (we fell over this before when you conflated $\R$ with $\R^1$). That is, an element of $\R^n$ is an ordered tuple which is notthe same thing as a vector. A vector is an ordered tuple with further structure added.

The initial definition of partial differentiation was fine, in that it operated upon an ordered tuple, which (in that context) is the most general onject you can start with.

If Larson is working directly with vectors then he's glossing over what (in pure mathematics) is an important detail. --prime mover 11:07, 3 April 2012 (EDT)


 * My Linear Algebra class gets to all the details of vector spaces in 2 weeks. If the problem isn't resolved by someone else by the time I finish learning it, I'll go over what I wrote and see if I can make my stuffs more precise. --GFauxPas 11:17, 3 April 2012 (EDT)


 * Okay so in the meantime I'll ignore all this, except for I'll put Partial Derivative back the way it was. --prime mover 11:41, 3 April 2012 (EDT)


 * Ignore it, fix it, flag it, or revert it as you like. Hopefully I'll be able to bring it up to PW standards by the end of linear algebra - but warning, it might not be for months :( I apologize if I should have waited until I had firmer footing before putting these pages up. --GFauxPas 12:45, 3 April 2012 (EDT)