Linear maps vs diffeomorphisms
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Linear maps vs diffeomorphisms
Approximate differentiability
Here is an interesting question on mathoverflow about approximate differentiability of a function. I use to think those two definitions are equivalent.
Definition: Let $f:E\to\mathbb{R}$ be a measurable function defined on a measurable set $E\subset\mathbb{R}^n$. We say that $f$ is approximately differentiable at $x\in E$ if there is a linear function $L:\mathbb{R}^n\to\mathbb{R}$ such that for any $\varepsilon>0$ the set $$ \{ y\in E:\, \frac{|f(y)-f(x)-L(y-x)|}{|y-x|} <\varepsilon \} $$ has $x$ as a density point.
Then this is the same as
A measurable function $f:E\to\mathbb{R}$ defined in a measurable set $E\subset\mathbb{R}^n$ is approximately differentiable at $x\in E$ if and only if there is a measurable set $E_x\subset E$ and a linear function $L:\mathbb{R}^n\to\mathbb{R}$ such that $x$ is a density point of $E_x$ and $$ \lim_{E_x\ni y\to x} \frac{|f(y)-f(x)-L(y-x)|}{|y-x|} = 0. $$
@adaf_gr #tiny #robot #meditating #on #singularity #with #projections #of #mathematical #objects which are not #defined, or a #point of an #exceptional #set where they #fail to be #well-behaved in some #particular #way, such as #differentiability (hier: Μέγαρο Μουσικής Αθηνών / Megaron - The Athens Concert Hall)
Continuity and Differentiability
Here is a continuous and differentiable piecewise defined function as described in this week's POTW. I used m = 2, n = 3, and g = 9.8 from values chosen by students in class today.
Differential - a story about being bad at English in Math class and it somehow matters [10/2/12]
So today in BC I raised my hand when Mr Scavo asked what the function was at that point. I said "It's non-differential?"
Scavo: Try again
Me: ...uhh...non-differentiality?
Scavo: ...One more try
Me: Ummmm... *at this point I feel like I'm on the wrong track here and I think Leart whispers 'differentiable' to me*
Scavo: Just add a b in there
Me: Differentia-BL (it sounds kinda weird because I really emphasized that B and the whole class laughs)
Scavo: Differentiable. It's just a matter of syntax blah blah
--later in the lesson, we do something similar with the same context--
Scavo: Meredith, would you like to try again?
Me: Differentiable! (I say it right at last LOL...as people laugh)
--after class--
Scavo: Should I ask you to say it again?
Me: What? Differentiable?
Scavo: Now you can say it! Why couldn't you say it before?
Me: I dunno, it was a weird word
Scavo: Well now you can say it. It should be as easy as saying dog
Me: Okay so when I see a dog I'll say "differentiability!"
--we both laugh and I leave followed by the usual 'bye Meredith'--
--Then I think of how dogs can be approached by two sides so they are differentiable and I think of how I should have said that but ah well, a few seconds too late--
hey again, cauchybro!
part 2 of complex analysis (toc)
skipping merrily along with complex analysis, we can basically gloss over a lot of results since they translate directly from real analysis (because the basic concept of a limit is still the same). For example, we define the limit of functions, continuity, and derivatives the same way as in real analysis. We get stuff like the product rule, derivative of polynomials, etc, as all the same.
Let's talk a little bit about derivatives of complex functions.