So the problem is that we have an equilateral triangle (side $\mathrm{a}$) with it's center in the origin that is rotating $\omega\hat{z}$. The magnetic field is in the $\hat{x}$ direction (not needed for my question) and I need to find the flux (trivial). The problem looks thusly: 
So, solving problems like that is not hard, but the hardest part is finding the correct parameterization for the $\vec{r}$ which describes our surface through which we are finding the flux. Now in the solutions the parameterization stated is the following:
$$\vec{r} = \frac{a}{2}\xi\hat{n}(t) + \frac{a}{2}\eta\hat{m}$$ where $\hat{m}$ describes the side that is at the $\pi/3$ with respect to $\hat{n}$ and $\xi,\eta$ are in range $[-1,1]$...so...
$$\hat{n}(t) = -\mathrm{sin}(\omega t)\hat{x} + \mathrm{cos}(\omega t)\hat{y}$$
This I understand.
The problematic part for me is the "vertical" part.
$$\hat{m} = -\frac{1}{2}\hat{n} + \frac{1\sqrt{3}}{2}\hat{z}$$
Now the solution is then simplified by introducing:
$$\xi = 2u-1\\
\eta = 2v-1$$
... and then when you go look for the surface differential (trivial) you derivate $\vec{r}$ with respect to $\mathrm{u}$ and $\mathrm{v}$ and get
$$\mathrm{d}^2r=[\frac{a^2\sqrt{3}}{2}[\mathrm{cos}(\omega t)\hat{x}+\mathrm{sin}(\omega t)\hat{y}]-\frac{a^2}{2}sin(\omega t)\hat{z}]\mathrm{d}u\mathrm{d}v$$
And finally, to find flux $\vec{B}\cdot \mathrm{d}\vec{A}$, one only has to realize that the area that describes the triangle is in:
$$u \in [0,1] \\
v \in [0,1-u]$$
I understand the $\hat{n}(t)$ but I can not figure out the $v \in [0,1-u]$ and $\hat{m}$
I do not quite understand the parametrisation given in the question, so I will give a parametrisation and work things out from there.
The typical way to parametrise a triangle with vertices given by position vectors $\vec{r}_1$, $\vec{r}_2$ and $\vec{r}_3$ is via
$$ p(\lambda, \mu) := \vec{r}_1 + \lambda(\vec{r}_2 - \vec{r}_1) + \mu (\vec{r}_3 - \vec{r}_1), \lambda, \mu \in [0,1] \text{ and } \lambda \leq \mu. $$
For all $t$, denote by $\vec{r}_1(t)$ be the position vector at time $t$ of the vertex of the triangle that initially has a negative $y$ coordinate, $\vec{r}_2(t)$ the (constant) position vector at time $t$ of the vertex that remains fixed under the rotation, and $r_3(t)$ the position vector at time $t$ of the vertex initially positioned with a positive $y$ coordinate.
Initially ($t = 0$), the position vectors of the vertices are given by
\begin{align} \vec{r}_1(0) &= \left(0, -\frac a 2, -\frac{\sqrt{3}}{6} a\right), \\ \vec{r}_2(0) &= \left(0, 0, \frac{\sqrt{3}}{3} a\right), \text{ and } \\ \vec{r}_3(0) &= \left(0, \frac a 2, -\frac{\sqrt{3}}{6} a\right). \end{align}
The positions of the vertices at time $t$ are given by rotating the initial positions by an angle $\omega t$ about the $z$-axis. Hence, for all $1 \leq i \leq 3$ and times $t$ we have $$\vec{r}_i(t) = R_z(\omega t) \vec{r}_i(0), $$ where $R_z( \omega t)$ is the matrix corresponding to a rotation about the $z$-axis by an angle $\omega t$. After calculating the $\vec{r}_i(t)$, we can give a parametrisation $p_t$ of the triangle at time $t$ using the method outlined at the beginning, which turns out to be $$ p_t(\lambda, \mu) = \frac{a}{2}\left(\lambda + 2 \mu - 1\right) \hat{n}(t) + \frac{\sqrt{3} a}{6} \left( 3 \lambda - 1 \right) \hat{z}, \lambda , \mu \in [0,1], \lambda \leq \mu.$$
Using the convention that the unit normal vector to the triangle is initially $\hat{x}$, the flux $\Phi(t)$ of the vector field $\vec{B}(x,y,z)$ through the triangle at time $t$ is equal to
$$ \int_0^1 \int_0^\mu \vec{B}\left(p_t(\lambda, \mu)\right) \cdot \left(\frac{\partial p_t}{\partial \mu} \times \frac{\partial p_t}{\partial \lambda} \right) d\lambda d\mu.$$ Running through the calculations, we get that
$$ \Phi(t) = \frac{\sqrt{3} a^2}{2} \int_0^1 \int_0^\mu \vec{B}\left(p_t(\lambda, \mu)\right) \cdot \left( \cos(\omega t) \hat{x} + \sin(\omega t) \hat{y} \right) d\lambda d\mu.$$
I apologise that I can't be of help interpreting the solution referred to in the question, but hopefully this gives you a gist of how to solve the problem.