The pores in a foam are allowed to overlap with each other. This isn't a sphere packing problem though. This is a question of the expected value of the volume these pores will occupy.
Homework Statement
This should be a pretty simple question but I can't find a straight answer in the literature. I want to simulate a 3D model of a metal foam by starting with an aluminum block and filling it with randomly placed spherical holes of constant volume. The foam should have a...
Homework Statement
The problem is to find the moments E(X^k) of f_x(x) = (\theta+1)(1-x)^\theta, 0 < x < 1, \theta > -1
Homework Equations
E(X^k)=\int_0^1 x^k (\theta+1)(1-x)^\theta dx
According to Mathematica, the solution is \frac{\Gamma(1+k)\Gamma(2+\theta)}{\Gamma(2+k+ \theta )}. I...
The problem is that the two boundary conditions
F(\omega)+G(-\omega)=0 and
F'(\omega)+G'(-\omega)=0
imply that F(\omega) and G(\omega) are constant. This does create an inconsistency.
Homework Statement
The problem is to solve
\phi_{yy}-c^2 \phi_{xx} = 0
\phi_y (x,0) = f'(x), x>0
\phi_x (0,y) = \phi(0,y) = 0, y>0 or y<0
Homework Equations
The solution, before applying boundary conditions is obviously
\phi(x,y)=F(x+c y)+G(x-cy)
The Attempt at a Solution
I start...
Homework Statement
You have a number system in base 10 with a precision of 5 digits. Which function is more accurate: x^2-y^2 or (x-y)(x+y)?
Homework Equations
None really.
The Attempt at a Solution
My intuition would tell me that (x-y)(x+y) is more accurate, since multiplication is...
Yeah, it might be. But your method works. Thank you so much for your help. You always overlook the simple solutions, right?
It turns out the answer is:
\frac{d\phi}{dr}=\frac{y-x y'}{(x+y' y)\sqrt{x^2+y^2}}
I also need to find \frac{d^2\phi}{d r^2}, but I think you just need to apply...
This isn't a homework question out of a textbook. But here's what I'm looking for:
Given variables phi, r, x, y such that
x = r sin(phi), y = r cos(phi)
and a function phi(r)
I need to find dphi/dr in terms of x, y, and dy/dx.
Homework Statement
Given Cartesian coordinates x, y, and polar coordinates r, phi, such that
r=\sqrt{x^2+y^2}, \phi = atan(x/y) or
x=r sin(\phi), y=r cos(\phi)
(yes, phi is defined differently then you're used to)
I need to find \frac{d\phi}{dr} in terms of \frac{dy}{dx}
Homework...
Well, that's not exactly right. But you can work a little index magic and come up with
\epsilon_{ijk}\epsilon_{pqr}=\left| \begin{matrix} \delta_{ip} & \delta_{iq} & \delta_{ir} \\ \delta_{jp} & \delta_{jq} & \delta_{jr} \\ \delta_{kp} & \delta_{kq} & \delta_{kr} \end{matrix} \right|