Recent content by Goklayeh

Maybe, I've solved: let y = x + v, for some (x, v) \in V, and consider any other point p \in M, together with a curve \gamma: (-\epsilon, \epsilon) \to M such that \gamma(0) = x, \dot{\gamma}(0) = v joining x, p. Without loss of generality, y = 0. If f(t):=\frac{1}{2}\left|\gamma(t)\right|^2...

Hello! Could anybody give me some hint with the following problem? Consider a smooth, compact embedded submanifold M = M^m\subset \mathbb{R}^n, and consider a tubular neighborhood U = E(V)\supset M, where E: (x, v) \in NM \mapsto x + v \in M is a diffeomorphism from a open subset of the normal...

Consider the unit ball B:=B_1(0)\subset \mathbb{R}^2. How can one prove that the set B\times B \setminus D, where D:=\left\{(x, x)\biggr| x \in B\right\} is the diagonal, is non-contractible? Is it even disconnected? Thank you in advance.

The equation is of the form \frac{\mathrm{d}y}{\mathrm{d}t} = P(t)y(t) + Q(t) with P(t)\equiv -1 and Q(t):=\sum_{n\ge 1}{\frac{\sin(nt)}{n^2}}. So, try with the formula y(t) =...

Hello everybody! I was looking for a counterexample to Gauss-Bonnet Theorem, that is, a region R \subset \Sigma (with \Sigma \subset \mathbb{R}^3 surface) such that \partial R isn't union of closed piecewise regular curves and for which the Gauss Bonnet Theorem doesn't holds, i.e. \iint_R{K...

Hello everybody! I have a really silly question concerning wave equation: consider the problem \left\{ \begin{matrix} u_{tt} &=& u_{xx} & x \in \mathbb{R}\\ u(x,0) &=& 0 & \\ u_t(x,0) &=& x(1-x)\chi_{\left[0,1\right]}(x)& \end{matrix} \right. the solution is given by d'Alembert's...

I was looking for a rigorous proof of this fact on my notes, but I think that is intuitively trivial (I was trying arguing by absurdum...). Maybe a look at this on a sphere could be useful to convince yourself (or, at least, I guess!)

Oops! I haven't seen your reply!

Ok, here is the proof. Recall that if \Sigma \subset \mathbb{R}^3 is a compact surface and \Pi is a plane, we can define the function "height from a plane" \begin{matrix} h:& \Sigma & \longrightarrow & \mathbb{R}\\ & p & \mapsto & \text{d}\left(p, \Pi\right) \:\:(\text{...

I'm saying that \Sigma \subset \mathbb{R}^3 compact \Rightarrow Gauss map restricted to \Sigma^+_0 is surjective and that's for sure! If this statement is the problem, I can post the proof. Otherwise, I didn't understand the objection! (Anyway, thanks for your time lavinia!)

I think I've solved: if \Sigma is a compact surface, then the Gauss map restricted to parabolic and elliptic points N: \Sigma^+_0:=\left\{p \in \Sigma\biggr| K(p) \ge 0\right\} \longrightarrow S^2 is surjective (this follows from a simple geometric argument), so \int_{\Sigma}{|K|} =...

Total "absolute" curvature of a compact surface Hi! Someone could help me resolving the following problem? Let \Sigma \subset \mathbb{R}^3 be a compact surface: show that \int_{\Sigma}{|K|\mathrm{d}\nu} \ge 4\pi where K is the gaussian curvature of \Sigma. The real point is that I want...

Consider a mechanical system (subject to gravity) consisting of an homogeneous bar, lying in the xy -plane, of mass m, length \ell and negligible section, with an extreme hinged in a fixed point P of the y axis. Now, suppose that the y-axis is rotating with angular velocity \omega = \omega(t)...

Could someone confirm or refute the following statement? f \in L^p\left(X, \mu\right) \: \Leftrightarrow \: \int_X{\lvert fg \rvert d\mu < \infty\: \forall g \in L^q\left(X, \mu\right) where 1<p<\infty,\: \frac{1}{p}+\frac{1}{q}=1 and (X, \mu) is a measurable space (of course, the...