Consider $\mathbb{R}^4$ with coordinates $(x,y,z,w)$. Let $M$ be a $2$-manifold in $\mathbb{R}^4$ parametrized by $\gamma(u,v)= \begin{bmatrix}u^2-v^2 \\ uv \\ u+v \\ u-v \end{bmatrix}, \ \ u^2+v^2 \leq 1$. Let $\omega=dx \wedge dy+dz \wedge dw$ be a 2-form. $$ \text{Then find } \int_M \omega.$$
I know that if $\omega=\sum a_{i_1, a_2,\ldots, a_k}(x) \, dx^{i_1} \wedge dx^{x_{i_2}} \wedge \cdots \wedge dx^{i_k} $ be a differential form and let $M$ be a $k$-manifold over which we wish to integrate, where $M$ has the parametrization $ M(u)=(x^1(u), x^2(u),\ldots, x^k(u))$ for $u$ in the parametrization domain $D$. Then the intgral is defind as $\int_M \omega = \int_D a_{i_1, i_2, i_3,\ldots, i_k} (M(u))\cdot J \ du^1 . du^2 \cdots du^k.$ where $J$ is the jacobian given by $J= \dfrac{\partial(x^{i_1} \cdots x^{i_k}) }{\partial(u^1\cdots u^k)}. $ But how to apply this formula to the given problem. I really can't . Please help me
So here are the details. Firstly we note that: \begin{align*} dx &= 2u du -2vdv\\ dy &= vdu + udv \\ dz &= du + dv\\ dw &= du -dv \end{align*}
Then we have that: \begin{align*} dx \wedge dy &= 2u^2 du \wedge dv - 2v^2 dv \wedge du\\ & = (2u^2+2v^2) du \wedge dv\\ dz \wedge dw &= -du \wedge dv + dv \wedge du = -2 du \wedge dv. \end{align*}
Now we have that: \begin{align*} \int_M \omega = \int_{u^2+v^2 \leq 1} (2u^2+2v^2 -2)dudv. \end{align*}
Details: During my calculations I've used that $du \wedge du = dv \wedge dv =0$, $du \wedge dv = - dv \wedge du$ and multilinearity.