Recent content by IridescentRain

create/caress/destroy

Ah, thanks for the advice. It's perfectly reasonable now that you've mentioned it, yet I would have probably gone and done just that --randomly select a latitude and a longitude-- without realising that the point distribution would then be denser near the poles. I'll be more careful than that...

Thanks for the suggestion. With a million points, that would make a very good approximation. (Obviously, the area of my region would be the sphere's area times the fraction of points inside the torus.) Cheers!

Sometimes (read: when you have some idea of what your solution should look like, at least qualitatively), you can rewrite your function as another function or your variable as another variable (the radial dependence of the spherical wave function in the separation-of-variables method comes to...

I remember I used Perl when I wanted to play around with strings a few years ago. C++ is more elegant, but back then I found Perl's handling of strings easier to work with (for the purposes of what I was trying to do, at any rate, which was comparing two strings and pointing out the differences...

Hi! Say I have a region described by any number of inequalities. This region is a surface in 3D space. How can I ask Mathematica to calculate the region's area? If it helps, my particular region is the intersection of a hollow sphere and a solid (i.e. filled-out) toroid-like surface. I'm...

Thanks, Millennial. I will.

Hello. I'm not terribly proficient with Bessel functions, but I know that those of the first kind are given by \begin{eqnarray} J_n(x) & = & \left(\frac{x}{2}\right)^n\,\sum_{\ell=0}^\infty\frac{(-1)^\ell}{\ell!\,\Gamma(n+\ell+1)}\,\left(\frac{x}{2}\right)^{2\ell}, \end{eqnarray} where...

But of course! So \left[k_\rho^2\frac{d^2R}{d(k_\rho\rho)^2}+\frac{k_\rho}{\rho}\frac{dR}{d (k_\rho \rho)}\right]\,\Phi\,Z-\left[\frac{n^2}{\rho^2}+k_z^2\right]\,R\,\Phi\,Z+\left(k_\rho^2+k_z^2\right)\,R\,\Phi\,Z=0;dividing by k+\rho^2\,\Phi\,Z...

Hey! Thanks for your help. All right, I have \phi(\vec{r})=R(\rho)\,\Phi(\varphi)\,Z(z),where R(\rho)=J_n(k_\rho\rho),\Phi(\varphi)=e^{in\varphi},Z(z)=e^{ik_zz}. Therefore, \frac{1}{\rho^2}\frac{\partial^2\phi}{\partial\varphi^2}=-\frac{n^2}{\rho^2}\phi,\frac{\partial^2\phi}{\partial...

Hello. I don't know how to prove that a certain function is a solution to the scalar wave equation in cylindrical coordinates. The scalar wave equation is \left(\nabla^2+k^2\right)\,\phi(\vec{r})=0,which in cylindrical coordinates is...

Ah, you're right. I misinterpreted the picture. Thanks. I'll check what the unit cell for the "cannonball arrangement" is and use that instead of the tetrahedron. I suppose I'll lose generalisability, though...

In fact, I can; the "cannonball arrangement" does just that. Take any four mutually adjacent spheres (which, of course, form a tetrahedron). Now take any three of those spheres and a new sphere which is mutually adjacent to all three; they will form a second tetrahedron which has a face in...

Hello. I wish to calculate the density of optimally packed spheres (i.e. the fraction of space which is occupied by the spheres). This number is approximately 0.74, but I know this because I have looked on the net, not because I have calculated it. I know the essentials of how to calculate...

All, right, I shall, thank you. =)