Smooth functions
I’ve been spending way too much time this week thinking about smooth functions. We covered the theorem that says a partition of unity exists for every compact subset of a Euclidean space. One of the homework problems was to extend this theorem to show that the partition functions could be chosen to be smooth. It wasn’t until I started trying to do the problem that I realized that the book doesn’t define what smooth means. For instance, if 
, what does it mean to say 
? I get the impression that that is equivalent to saying
exist for all 
. But what does this mean really? I can think of differentiation in terms of linear algebra (matrices, really) up to the Hessian, but what happens when you try to define e.g. 
? Most books I’ve flipped through either only define smooth functions on 
, the obvious case, or in Banach spaces, which I don’t understand how to apply to my situation. I’m still looking for a reasonable motivation for my intuitive understanding of smooth.
Possibly relevant posts:
- The implicit and inverse function theorems (9/12/2006)
- Differential of the determinant (6/29/2007)
- Interpolating splines (1/22/2006)
Honestly, to get a good handle on differentiation in R^n you need to stop thinking about the matrix representation
and think of the derivative as a mapping from a p-form to a p+1-form. (which then makes vector calculus
easy, since once you understand differential forms, stokes, et al follow). I recommened Munkres’ Analysis on Manifolds
for a good description of the d operator.
I think smooth means C^infinity or in practice “As many derivatives I need in my proof/algorithm, I have available to me”.
In that connection, I was flipping through Rudin today (as opposed to actually reading it), and I saw that the higher-dimensional Taylor theorem might supply a reason for that definition of
.
Thanks for the reference to Munkres. I looked it up in the UH library but couldn’t find it. Time for interlibrary loans.