Exterior derivative of $\omega=\int_M(y^2-x^2)dy\wedge dz+(z-x)dz\wedge dx+(2z-y)dx\wedge dy$

Question

I want to find exterior derivative of $\omega=(y^2-x^2)dy\wedge dz+(z-x)dz\wedge dx+(2xz-y)dx\wedge dy$ and I am not sure if I do that correctly.

I start with $$\omega=(-2xdx+2ydy)\wedge dy\wedge dz+(dz-dx)\wedge dz\wedge dx+(2zdx-dy+2xdz)\wedge dx\wedge dy=-2xdx\wedge dy\wedge dz+2xdz\wedge dx\wedge dy=-2xdx\wedge dy\wedge dz+2xdx\wedge dy\wedge dz=0$$

Did I make something wrong?

Answer 1

It is well; although in the integral you have put the last term different. But the exterior derivative of $\omega=(y^2-x^2)dy\wedge dz+(z-x)dz\wedge dx+(2xz-y)dx\wedge dy$ is correct.