Wedge product

Definition for covectors ...

Definition for differential forms... See @baez1994gauge page 59.

Proposition (anticommutativity)
If $f \in A_{k} (V)$ and $g \in A_{l} (V)$ then $f \land g = (- 1)^{k l} g \land f$ .
$◼$

Interpretation: see visualization of k-forms.

Important fact: Wedge product of two 1-forms:

α \land β = \sum_{1 \leq i < j \leq n} (α_{i} β_{j} - β_{i} α_{j}) d x^{i} \land d x^{j}

So the wedge product of two 1-forms is zero if and only if they are proportional: $α \land β = 0$ if and only if $α = μ β$ for a non vanishing smooth function $μ$ . Important for the resolution of Pfaffian equations.

It is also deduced from the anticommutativity:

α \land α = - α \land α ⟹ α \land α = 0

Visualization

See visualization of k-forms