Recent content by quasar987

This is just linearity of the expectation. You are assuming X and Y have expectation 0 and are independent. Develop (X+Y)^3, use linearity of E[.], then use independence and centrality to get E[X^2Y] = E[X^2]E[Y]=0 and E[XY^2] = E[X]E[Y^2]=0.

Are you saying the area under the curve of the positive function $f(x)=x^2$ is 0? I assign to the variance of a function $f$ the same meaning that Var[X] hold for a random variable X; it is a measure of dispersion: small variance means f tends to sheepishly sticks close to its mean µ[f] whereas...

I'm not saying the term "functional mean" is well known, but that the concept of the mean of a function is well known. And I am curious if the corresponding concept of "functional variance" (aka variance of a function) appears somewhere in the literature.

For a continuous function f:[a,b]-->R there is a well-known notion of "functional mean" $$ \mu[f] = \frac{1}{b-a}\int_a^bf(x)\,dx. $$ I am wondering if there is a name for the corresponding notion of "functional variance", i.e. the quantity $$ \frac{1}{b-a}\int_a^b(f(x)-\mu[f])^2\,dx. $$ ...

To extrapolate a bit on e_jane's answer... I was watching recently the RI talk with John Stillwell () and he was saying that for the LONGEST TIME people were not working with symbols. These started being used as we do today only around the year 1600 (!) On the other hand, the square completion...

Thanks for clarifying. My post was just a guess because I don't know what is meant by "the complete boundary" of a surface or of a part of it.

I have not read this text but I think I remember reading about it that Riemann is basically taking about Betti numbers but he does it in this language of "k-connectivity". The basic idea I think is this: take a sphere and take any closed loop a on it and take the scisors to it. This then splits...

I think we might need to add the condition that a is smooth here. For ##\Leftarrow## we assume that ##a## has the property that a function ##f:U\rightarrow \mathbb{R}## is smooth iff the composition ##f\circ a^{-1}## is smooth. From this hypothesis, we want to show that ##b\circ a^{-1}## and...

I think you basically answered yourself there. There is a manifold, and an atlas on it. So that's a collection of local homeomorphisms into euclidean space such that their composition on overlaps are diffeomorphisms. Such a system allows one to define smoothness of a function...

You have the right idea for "==>" but again, sorry to say but your post is a mess. 1) Very first line, a and b are chart on the same manifold so they should both map into R^m. 2) In ">2." you give the definition of a smooth map between manifolds. Yet this question is purely about functions...

This is an absolute notational disaster. If you want to continue the discussion we need to start from scratch, use good notation and stick with it. But honestly, I gave the complete solution in post #10. Have you tried to understand it?

In post #8 you introduced the symbols ##a## and ##b## but I just assumed you made a mistake and you meant ##a=\phi##, ##b=\psi##. But now you are seemingly purposefully making a distinction between ##a##, ##b##, ##\phi## and ##\psi##. What is going on?! If you really thought of ##a## and ##b##...

That is indeed the question. Working from the hypothesis that the functions ##\psi^1\circ f,\ldots, \psi^n\circ f## are smooth on ##V_p##, can you conclude that ##f## is smooth on ##V_p## ? It is completely tautological. Meaning that just by unraveling the definitions we can see that these are...

What does it mean to say that the functions ##\phi^j\circ f## are smooth ? What does it mean to say that the function ##f## is smooth at ##p\in M## ?

In general, a smooth manifold is not a subset of euclidean space. It is a topological space that's locally homeomorphic to euclidean space and such that, on overlapping charts, the transition functions are smooth. Among the manifolds defined in this general way, there are some that occur as...