Consider the differential forms on $\mathbb{R}^3$,

$\omega_1 = xy \space dx + z \space dy + dz$ , $\omega_2 = x \space dy + z \space dz$.

I need to determine $\omega_1 \wedge \omega_2$.

However, I do not know how to find such wedge products. Any help is appreciated.


$$ \sum_I a_I dx^I \wedge \sum_J b_J dx^J: = \sum_{I, J} (a_I b_J)\ dx^I \wedge dx^J$$


$$(x dx + y dy) \wedge (2 dx - dy) = 2x \ dx \wedge dx- x \ dx \wedge dy + 2y \ dy \wedge dx- y \ dy \wedge dy\\ \hspace{-.41in}= (-x-2y)\ dx \wedge dy$$


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