The skyrmion number is defined as $$n=\frac{1}{4\pi}\int\mathbf{M}\cdot\left(\frac{\partial\mathbf{M}}{\partial x}\times\frac{\partial\mathbf{M}}{\partial y}\right)dxdy$$ where $n$ is the topological index, $\mathbf {M}$ is the unit vector in the direction of the local magnetization within the magnetic thin, ultra-thin or bulk film, and the integral is taken over a two dimensional space.
It is known that $\mathbf{r}=\left(r\cos\alpha,r\sin\alpha\right)$ and $\mathbf{m}=\left(\cos\phi \sin\theta,\sin\phi \sin\theta,\cos\theta\right)$. In skyrmion configurations the spatial dependence of the magnetisation can be simplified by setting the perpendicular magnetic variable independent of the in-plane angle ($ \theta \left(r\right)$) and the in-plane magnetic variable independent of the radius ($ \phi \left(\alpha\right)$). Then the skyrmion number reads: $$n=\frac{1}{4\pi}\int_0^\infty dr\int_0^{2\pi}d\alpha\ \frac{d\theta\left(r\right)}{dr}\frac{d\phi\left(\alpha\right)}{d\alpha}\sin\theta\left(r\right)=\frac{1}{4\pi}\ [\cos\theta\left(r\right)]_{\theta\left(r=0\right)}^{\theta\left(r=\infty\right)}[\phi\left(\alpha\right)]_{\theta\left(\alpha=0\right)}^{\theta\left(\alpha=2\pi\right)}$$
My question is: is $\frac{\partial\mathbf{M}}{\partial x}\times \frac{\partial\mathbf{M}}{\partial y}$ a curl product and what is the output of this term? How to reach to the final equation then?