Recent content by cacofolius

1. Two particles' spin Hamiltonian

Homework Statement Hi, I'm trying to familiarize with the bra-ket notation and quantum mechanics. I have to find the hamiltonian's eigenvalues and eigenstates. ##H=(S_{1z}+S_{2z})+S_{1x}S_{2x}## Homework Equations ##S_{z} \vert+\rangle =\hbar/2\vert+\rangle## ##S_{z}\vert-\rangle...

3. Cylinder carrying volume and surface current, H field

Homework Statement We have an infinite cylinder that, from radius 0 to a, has a volume current density ##\vec{J(r)}=J_{0}(r/a) \hat{z}## , then from a to 2a, it has a material with uniform linear magnetic permeability ##\mu=(3/2)\mu_0## , and at the surface, it has surface current...
4. 2 True/False Questions -- Integral and convergence

Thanks pasmith, indeed that is a counterexample, but I have no idea how to define that function (except for the x=0, when x<0, part, of course). Nevertheless, describing it as you did would not count as correct answer (whatever the explicit expression for its formula might be) in replying that...
5. 2 True/False Questions -- Integral and convergence

The b) is a different problem, that says if the integral from zero to infinity converges then its integrand must go to zero when x extends to infinity. I was refering to that problem when I brought up the series. For the a) problem, following your suggestion: suppose that for some ##a \subset...
6. 2 True/False Questions -- Integral and convergence

Thank you for your patience, I misread you, I see what you mean now, I'll work on that. Meanwhile I thought something for the b) part. If could treat the problem (since f is continous) from a series point of view (maybe justifying this with the archimedean property of real numbers that states...
7. 2 True/False Questions -- Integral and convergence

Thanks for your reply Hallsoflvy, but I don't quite understand what you are saying. The statement has the condition that the integral is zero on every interval ##[a,b] \subset [0,1]##, therefore how could I suppose that for some ##a## the conclusion is that its integral is not zero? I thought of...
8. 2 True/False Questions -- Integral and convergence

Homework Statement a) If ##f: [0,1] \rightarrow \mathbb{R}## is continous and ##\int^{b}_{a} f(x)dx = 0## for every interval ##[a,b] \subset [0,1]##, then ##f(x)=0 \forall x \in [0,1]## b) Let ##f: [0,\infty) \rightarrow [0,\infty)## be continous. If ##\int^{\infty}_{0} f(x)dx## converges...
9. Differentiability of a function -- question on bounding

Thanks Ray, I understand now what you meant.
10. Differentiability of a function -- question on bounding

Hi Ray Vickson, I was trying to use the theorem that states that if I can decompose the function like this ##F(r,\theta)=H(r)G(\theta)## and ##\lim_{(r) \rightarrow (0)} H(r) = 0## and ##G(\theta)## is bounded in [0,2pi] then the original function is differentiable, but I see now that since the...
11. Differentiability of a function -- question on bounding

I'm sorry, that happens for hurrying. I forgot to divide by ##||(u, v)||## in the definition, which gives me an extra ##r## in the denominator: ##\frac {r^3 cos(\theta) sin(\theta)^2} {r^3} = cos(\theta) sin(\theta)^2## which means it doesn't exist, therefore is not differentiable in (0,0)...
12. Differentiability of a function -- question on bounding

You're right, Samy_A, it doesn't, and with the polar coordinates I have a function which is the product of two functions ##H(r,\theta)=F(r)G(\theta)## and while ##\lim_{(r) \rightarrow (0)} F(r) = 0##, ##G(\theta)## is bounded in [0,2pi], therefore the original function is differentiable in...
13. Differentiability of a function -- question on bounding

Thanks for the help!
14. Differentiability of a function -- question on bounding

Homework Statement I need to see if the function defined as ##f(x,y) = \left\{ \begin{array}{lr} \frac{xy^2}{x^2 + y^2} & (x,y)\neq{}(0,0)\\ 0 & (x,y)=(0,0) \end{array} \right.## is differentiable at (0,0) Homework Equations [/B] A function is differentiable at a...
15. Hoop rolling on cylinder without slipping

Hi OldEngr63, thanks for your reply, I tried your approach with the angles, so I got ##\hat{θ}) 2m(R+a) \ddot{θ} + mg sinθ=0 ## which is nice for small angle approximation and describing the oscillatory movement. And Fr meant friction force. Sorry it took long for me to answer back. Thanks...