Let $\omega$ the volume $2$-form over $S^1 \times S^1$, find: $$\int_{S_1 \times S_1} \omega$$
Let $I=[0, 2\pi]$. I know that a parametriaztion of $S^1$ is $\phi(\theta)=(\cos(\theta),\sin(\theta)), \theta \in I$
Then a parametrization of $S_1 \times S_1$ is $F:I \times I \rightarrow S_1 \times S_1$ defined as $F(\theta,\varphi)=(\phi(\theta),\phi(\varphi))$. Then i need calculate: $$\int_{S_1 \times S_1} \omega=\int_{I\times I}F^*(\omega)$$
I'm stuck here, i don't know how to calculate the pullback $F^*(\omega)$, i need charts for this?