I have to calculate the integral $\int_{\gamma}d\omega$ $$\omega=(y+z)dx+(z+x) dy+(x-y) dz$$ $\gamma$ is the intersection between the sferic surface $x^2+y^2+z^2=1$ and che plane $y=z$
I want to solve it computing $d\omega$ directly with wedge products, so I found $d\omega=-2dy\wedge dz$. Now I establish a parametrization for $\gamma$ which is $$\begin{cases} x=cost\\ y=\frac{1}{\sqrt{2}}sint\\ z=\frac{1}{\sqrt{2}}sint \end{cases} $$ Where $t\in[0,2\pi]$, so I have to evaluate this integral: $$\int_{[0,2\pi]} -2dy\wedge dz(-sint, \frac{1}{\sqrt{2}}cost, \frac{1}{\sqrt{2}}cost) $$
But I don't know how to evaluate that, usually I use this method when integrating a 2-form on a 2-surface. What should I do?
You wish to apply Stokes theorem
$$ \int _{\gamma} \omega = \int_{D} d\omega,$$
where $D$ is a surface with $\partial D = \gamma$. You need to find this $D$ first.
There are lots of them, the simplest one is the intersection of the plane $y=z$ with the unit ball. It has a parametrization
\begin{cases} x= r \cos t\\ y=\frac{r}{\sqrt{2}}\sin t\\ z=\frac{r}{\sqrt{2}}\sin t \end{cases}
with $0<r<1$ and $ 0<t<2\pi$.
Or you can just use your parametrization of $\gamma$ and integrate $\omega$.