Wedge product

Definition for covectors ...

Definition for differential forms...

Proposition (anticommutativity)

If $f \in A_k(V)$ and $g \in A_l(V)$ then $f \wedge g = (-1)^{kl} g \wedge f$.

$\blacksquare$

Interpretation: see visualization of k-forms.

Important fact: Wedge product of two 1-forms:

$$ \alpha\wedge\beta = \sum_{1\le iSo the wedge product of two 1-forms is zero if and only if they are proportional: $\alpha\wedge \beta=0$ if and only if $\alpha=\mu \beta$ for a non vanishing smooth function $\mu$. Important for the resolution of Pfaffian equations.

It is also deduced from the anticommutativity:

$$ \alpha \wedge \alpha=-\alpha\wedge \alpha \implies \alpha \wedge \alpha=0 $$

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es


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