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Dec
22
comment Does $(x, y) \to (e^x \cos y, e^x \sin y)$ map open sets to open sets
thanks - see my comment to cactus314's answer
Dec
22
comment Does $(x, y) \to (e^x \cos y, e^x \sin y)$ map open sets to open sets
thanks, I did see this after searching for the question, but this is a question from my real multivariable calculus course (no complex analysis) so I'm sure there's a way to do this without proving that analytic functions map open sets to open sets
Dec
22
awarded  Student
Dec
22
comment Does $(x, y) \to (e^x \cos y, e^x \sin y)$ map open sets to open sets
@peterag - yes, I think that works. I am kicking myself for not seeing this. If you write is as an answer I'll accept it
Dec
22
comment Does $(x, y) \to (e^x \cos y, e^x \sin y)$ map open sets to open sets
This is a basic multivariable calculus course, so I haven't encountered diffeomorphisms yet, but I'll look into it
Dec
22
asked Does $(x, y) \to (e^x \cos y, e^x \sin y)$ map open sets to open sets
Feb
20
comment What's your favorite proof accessible to a general audience?
Another reason I like it - the more biased the coin is, the longer it takes to get one random bit out
Sep
30
awarded  Explainer
Jan
19
answered Mathematically, why does $[ma]\mathrm{d}x = [mv]\mathrm{d}v$?
Sep
25
revised Why is integral of $(\tan x)^3$ not $((\sec x)^2)/2 - \ln(\sec x)$?
add latex formatting
Sep
25
suggested approved edit on Why is integral of $(\tan x)^3$ not $((\sec x)^2)/2 - \ln(\sec x)$?
Sep
24
comment L'Hopsital Rule Understanding
let us continue this discussion in chat
Sep
24
comment L'Hopsital Rule Understanding
That's not true, $f'/g'$ might be undefined while $\lim f'/g'$ exists.
Sep
24
comment L'Hopsital Rule Understanding
Fair enough. But why doesn't your answer point out this limitation of your proof? I can comment on your answer instead if you want.
Sep
24
comment L'Hopsital Rule Understanding
I think any proof or proof intuition for l'Hopital's rule must be roughly as complex as mine. Your proof shows that $\lim f/g = f'/g'$, but l'Hopital's rule is that $\lim f/g = \lim f'/g'$. The later is much harder to prove than the former. In fact, I think you should point out this limitation in your answer
Sep
24
comment L'Hopsital Rule Understanding
Your proof doesn't work for applying l'Hopital's rule twice, your final expression is a ratio of derivatives without a limit in front.
Sep
24
comment L'Hopsital Rule Understanding
Because if you apply l'Hopital's rule twice, you are eliminating the zeroth and first order terms, and focusing on the quadratic terms of the functions
Sep
24
comment L'Hopsital Rule Understanding
then you need to talk about quadratic, cubic, quartic...etc approximations. I know the proof of taylor's theorem isn't very simple, but I find the fact that any nice function has a first, second, etc order approximation to be self-evident
Sep
24
comment L'Hopsital Rule Understanding
I know, this is how I express the intuition that "the first-order term determines the limit" which I think of when using l'Hopital's rule.
Sep
24
answered L'Hopsital Rule Understanding