Culture & Recreation

What's the opposite of a cross product?

So we have:

$$
\langle x_1, y_1, z_1\rangle\times\langle x_2, y_2, z_2\rangle = \langle y_1z_2 - z_1y_2, z_1x_2 - x_1z_2, x_1y_2 - y_1x_2\rangle
$$

Then we have:

\begin{align}
\langle x_3, y_3, z_3\rangle\times\left(\langle x_1, y_1, z_1\rangle\times\langle x_2, y_2, z_2\rangle\right) = & \langle y_3(x_1y_2 - y_1x_2) - z_3(z_1x_2 - x_1z_2),\\
&z_3(y_1z_2 - z_1y_2) - x_3(x_1y_2 - y_1x_2),\\
& x_3(z_1x_2 - x_1z_2) - y_3(y_1z_2 - z_1y_2)\rangle
\end{align}

Now this looks very complicated (and it is--if you need to solve it "as is").Then this becomes:

\begin{align}
\langle x_3, y_3, z_3\rangle\times\vec{n}' = & \langle y_3n'_z - z_3n'_y, z_3n'_x - x_3n'_z, x_3n'_y - y_3n'_x\rangle
\end{align}

Now we are trying to find $\langle x_3, y_3, z_3\rangle$ which is the inverse.

Verify integrals with residue theorem

Let
$$\begin{equation*}
f(z)=\frac{e^{iaz}}{\left( 1+z^{2}\right) ^{2}}=\frac{e^{iaz}}{
(z-i)^{2}(z+i)^{2}}.\end{equation*}\tag{1}
$$
The residue of $f(z)$ at $z=i$ is
$$
\begin{eqnarray*}
\underset{z=i}{\mathrm{res}}f(z) &=&\frac{1}{(2-1)!

How do you parameterize a sphere so that there are “6 faces”?

I'm trying to parameterize a sphere so it has 6 faces of equal area, like this:



But this is the closest I can get (simply jumping $\frac{\pi}{2}$ in $\phi$ azimuth angle for each "slice").Parametrizing a square with a square is much easier:

up = NestList[RotateLeft, {s, t, 1}/Sqrt[s^2 + t^2 + 1], 2];
ParametricPlot3D[
Join[up, -up],
{s, -0.

Orbit vs. Cycle

An orbit is a set, a cycle is a permutation of a set (which permutes its elements cyclically)


Reference: Jamie Mulholland http://www.On the other side, given a group acting on some set $X$, then the orbit of some $\alpha \in X$ is
$$
\alpha^G := \{ \alpha^g : g \in G \}
$$
i.

Question regarding Goursat's Theorem

I read through and pretty much understand most of Goursat's Theorem in $\textit{Complex Analysis}$ by Gamelin.The theorem states that if $f(z)$ is a complex-valued function on a domain $D$ such that $f'(z_0)$ exists at each point $z_0 \in D$, then $f(z)$ is analytic on $D$.

Identifying maximal, greatest elements on a Hasse/lattice diagram?

So, while I understand the difference between maximal elements and greatest elements, I'm having trouble understanding how to identify them on a lattice diagram (just as an example, the lattice diagram of the power set of A = {1,2,3}).From what I understand (and this could be wrong), the element {1,2,3} of the power set would be both the greatest element (since {1,2,3} is drawn at least as high as every other element in the diagram) and maximal element (since nothing is drawn higher than {1,2,3}).

Complex Numbers with the prey-predator equilibrium?

One possibility is $x=\frac{-h}{\alpha}, y=0$ and another one is $x=\frac{\gamma}{\rho}, y = \frac{\alpha}{\beta}+\frac{\rho h}{\beta\gamma}$ (plug these values in to see it for yourself).) neighborhood of any point in the state space, the nonlinear system $\dot x = f(x)$ behaves like a linear system $\dot x = Ax$.