Let $M$ be embedded in (R^{n+1},g), to calculate curvature on $M$, do we use $g$ or $g_M$? Let's say $M$ is a sphere (radius = 1/2) with n = 2. Let's say something easy like g = (x^2 + y^2)g_\mathbb{R^{3}} where $g_{\mathbb{R}^3}$ is standard metric. Do we calculate using g = (1/2)^2g_{\mathbb{R}^3}?
btw, I pinged you with this on disc a while ago, but you're gone now, so here it goes again: do you know that any two involutions with the same (finite) number of fixed points on an orientable surface are conjugate by a diffeomorphism?
Literally my example, let's say (R^3, (x^2 + y^2)g_{R^3}), if I look at M = S^2(1/2)embedded in R^3. To calculate curvature on S^2, do I use g = (1/2)^2g_{R^3} or keep it at g = (x^2 + y^2)g_{R^3}?
why wouldn't it change the curvature? Aren't I scaling everything by 1/4? i m guessing what i said is true? so if I change the sub manifold to another surface like x^2 + y^2 + 2z = 2^2, and i change the metric of the ambient space to something convenient like g = (x^2 + y^2 + 2z^2)g_{} = 2^2 g_{}
my question is inspired by the examples done in Peterson's page 3 of the book for context.
he also talks about induced metrics after embedding
a lot of diagrams on the internet explaining port and starboard lights on planes use a figure depicting the plane as viewed from above. you know, the most common way you see planes in real life.
sometimes there isn't enough in the image alnoe to make clear that it's a top view. that's funny to me. i wish we lived in a nonorientable universe.
jetblue had basically been at war with the LGB authority for about ten years. constantly violating noise regulations and sitting on a bunch of unused flight slots to keep southwest out. covid was the last straw.
i had a flight home canceled in apr 2020 and made the mistake of getting voucher for my $900 instead of a refund. now there is a change the airline will go belly up.
my wife's uncle has a company that makes very specialized electronics. i keep telling him he needs to get some IP going, sue people. he says that isn't how it works in his industry.
:-). our company used to get threats, we responded to the investors of the threatening company indicating our willingness to go legal to protect our good name.
silly stuff like 'you stole our idea'. when they had no idea of the tech incolved.
that's my favorite. 'you stole our idea' when they have no idea how it works, or surprisingly commonly, no idea when it was first created and on the market.
one of this guy's letters was like that. "your product is infringing my patent filed in 2007 and issued in 2012." the product had been on the market since 2001.
with closely held companies, or companies that have gone through a chain of acquisitions or rebranding, it can sometimes take a lot of work to figure out when a product was first released. but that is definitely job 1 of suing someone on a patent. make sure they weren't selling the thing that you say is your invention before you came up with it.
i saw it in the theater when it came out. the torture device where they take years off of cary elwes's life was a little intense for a 7 year old. i still loved the movie.
its funny, in ireland many years ago it was discovered that a bishop (casey) had a grown up son in new york. however, people did not really care about that, but what incensed the populace was the fact that he basically disowned his son. start of a long downward slide.
maybe for the first few days the rep reward goes if you find a dup, but after a few days answers get the rep :-). if you are a rep hunter like myself, of course.
@copper.hat I think that something a bit similar already is in place. IIRC, if you have answer of score at least 3 and it remains visible after 90 days, you keep the rep even if the answer is deleted.
I am not really sure what to respond to this. We would be going through the stuff that was chewed over many many times.
TBH it was a bit unclear what you mean, when I read your wording: "maybe for the first few days the rep reward goes if you find a dup, but after a few days answers get the rep."
might space f too far from the matrix to be function application. a free standing x over y, like \frac{x}{y} but without the bar in between them, is probably what you want
my daughter had a great time on the balance beam at the park today. she could climb onto the highest level of it all by herself, stand up, and walk down the levels, and walk up again. she fell off a few times but got right back on. i did nothing to help her. if anyone asks, it's because i want her to learn self reliance, but i was mostly just tired and wanted to sit down.
we'll probably start with jackie brown. she was born almost across the street from the del amo mall. i got food in the food court where they did the money handoff while waiting for her to be born.
there's some nightmare story of someone developing an elaborate theory of some class of functions only to learn at their defense or soon after that the only functions in the class are constant. urban legend, i'm sure.
my main doubt is when you substitute it with K(b-a)^2 just for understanding what its magnitude is for the left part you get K (ne)^2 but for right part you get n(e)^2
no, i accidentally linked to one you had already put above, i glanced at it and i thought i saw something different. it was too late by the time i realised to delete it
:-)
it doesn't help that my typing is so awful
actually, it is good that many are closed and link to the above one.
But I did not think of using it in this case either. (I would guess real-analysis, not holder-spaces, as the choice of the tag when searching for this.)
generally when i am trying to find a result for myself, i can find it quickly, but when i am searching for a dup, i am less successful. i do not understand why.
maybe there are more constraints when looking for a dup. who knows.
anyway, thanks for the tag suggestion :-). i'm going to sleep shortly!
@MartinSleziak approach 0 was really useful. @copper.hat yes differentiation method is easier to do the problem, and the please evaluate this (|\sum_{k=0}^{n-1}(f(x_{k+1})-f(x_k))|) given in
Let us take two arbitrary real numbers $x$ and $y$. Let $h=y-x.$
Let us subdivide $(x,y)$ into $n$ equal length intervals $(x_k,x_{k+1})$ with $x_0=x$ and $x_n=y$, (thus all with length $\tfrac{h}{n}$):
$$|f(x)-f(y)|=|\sum_{k=0}^{n-1}(f(x_{k+1})-f(x_k))|\leq \sum_{k=0}^{n-1}|(f(x_{k+1})-f(x_k)|...
a number of high school level books implicitly advance this proposition. in the background, there is the fact that a number "is" a linear map on a one-dimensional vector space.
the problem with single variable calculus is that too many things are synonymous with other things because the dimension is so small that a number of different actors have to play the same roles.
composition of linear maps is multiplication. it is literally multiplication in the one dimensional case. in higher dimensions it is matrix multiplication.
some beginning calculus books introduce the derivative as a linear map when they discuss differentials. if y = f(x) then dy = f'(x) dx. fixing a value of x, you take dx, presumed to be independent of x, as a variable. it is multiplied in linear fashion by the fixed number f'(x). the result is a linear function of the variable dx.
Let us take two arbitrary real numbers $x$ and $y$. Let $h=y-x.$
Let us subdivide $(x,y)$ into $n$ equal length intervals $(x_k,x_{k+1})$ with $x_0=x$ and $x_n=y$, (thus all with length $\tfrac{h}{n}$):
$$|f(x)-f(y)|=|\sum_{k=0}^{n-1}(f(x_{k+1})-f(x_k))|\leq \sum_{k=0}^{n-1}|(f(x_{k+1})-f(x_k)|...
