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Can somebody explain me step by step how can I compute the wedge product $X\wedge Y$ of two vector fields, $X,Y$, in $\mathbb{C}^2$?

We can consider $$ X=X_1\partial_x+X_2\partial_y $$ and $$ Y=Y_1\partial_x+Y_2\partial_y $$


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I take it $x$ and $y$ are complex coordinates? – Muphrid May 14 '13 at 20:25
Yes @Muphrid they are complex – OhMyGod May 14 '13 at 20:25

Expand out. $ \ \ e_i \wedge e_i=0, \ \ e_i \wedge e_j=-e_j \wedge e_i$

$e_1=\partial_x, e_2=\partial_y \\X\wedge Y=X_1Y_1e_1\wedge e_1+X_1Y_2e_1\wedge e_2+X_2Y_1e_2\wedge e_1+X_2Y_2e_2\wedge e_2\\ =(X_1Y_2-X_2Y_1)e_1\wedge e_2=(X_1Y_2-X_2Y_1) \partial_x \wedge \partial_y$

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What does the objecct $\partial_x\wedge\partial_y$ means? – OhMyGod May 14 '13 at 22:57

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