Recent content by daishin

Sorry. It's not obvious. It seems continuous though. I know for given epsilon > 0, I need to find delta>0 such that ||f||<delta implies ||Af||<epsilon. Hmm.. How can I find such delta? Or use cauchy sequence?

Homework Statement Let 1\leqp\leq\infty and suppose (\alpha_{ij} is a matrix such that (Af)(i)=\sum^{\infty}_{j=1}\alpha_{ij}f(j) defines an element Af of l^{p} for every f in l^{p}. Show that A is a bounded linear functional on l^{p} Homework Equations Isn't this obvious if we apply...

I think we can always find globally defined 1-forms w_1, w_2,...w_n on M which in some small neighborhood B, they span M* for each p in B. If not please correct me. My question came from the proof of Frobenius integrability theorem in Spivak Volume 1. It is a chapter 7 Theorem 14. He starts the...

Let M be a smooth manifold. Locally we can choose 1-forms \omega^{1},\omega^{2},...\omega^{n} whish span M^{*}_{q} for each q. Then are there vector fields X_{1}, X_{2}, ...,X_{n} with \omega^{i}(X_{j})=\delta^{i}_{j}? Here \delta^{i}_{j} is Kronecker delta. By vector fields, I meant vector...

Sorry what I meant was: Let w be a 1-form on smooth manifold M. Let f be a continuous function from M to R. Then is there a vector field X on M such that w(X)=f in some neighborhood of p in M?

Let w be a 1-form on smooth manifold M. Then is there a vector field X such that locally w(X)=f where f:M-->R continuous? How can I prove it? Thanks.

Morse Could you explain why if a linear combination of squares has both signs, then it can't be an extreme point? Hessian is a matrix of second partial derivatives. I suppose you are saying that: If the hessian is positive definite(i.e all the eigenvalues are positive)at some point p...

I started reading a Morse theory by Milnor and am not understanding something. I am reading the proof of Theorem on page 25: Let M be a compact manifold and f be a differentiable function on M with only two critical points, both of which are non-degenerate, then M is homeomorphic to a sphere...

streographic projection Using stereographic projection,(say (s,t)) I attained follwoing condition if I assume xdydz is exact on S^2, for some smooth function g and f, -df/dt+dg/ds = (-24(s^2)(t^2)-8(s^4)-8s)/(1+s^2+t^2)^4. Now in order to show xdydz is not exact, it suffices to show such...

Homework Statement Let w be the form w= xdydz in R^3. Let S^2 be the unit sphere in R^3. If we restrict w on S^2, is w exact? Homework Equations The Attempt at a Solution My guess is w is not exact on S^2. Suppose w is exact on S^2. Then w=da for some 1-form a=fdx+gdy+hdz...

I shall come back tomorrow. Any help would be appreciated!

f* is a pull back if f is defind from M to N then there is a map f* from the differential forms on N to a differential forms on M

I don't understand why the proof of integral(df)=integral(f'(x)dx) is related to my question. Could you explain more?

Why?? Could you provide an example of f such that the integra of f*w is not equal to the integral of w?? By the way, do you know what f* means??

So although initially(when we just looked at the problem) we don't know what f is, the problem is well defined. I don't think we can determine what function f is even after when we solve the problem. But I am not sure.