exterior product

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Define the exterior product or wedge product between two differential forms \eta\, and \mu\, of degrees p\, and q\, respectively as the p+q\,-form denoted by

\eta \wedge \mu\,,

with the following properties:

From

  • (\eta_1 + \eta_2)\wedge(\eta_1 + \eta_2) = \eta_1\wedge\eta_2 + \eta_2\wedge\eta_1 = 0\,,

the property

  • \eta \wedge \mu = (-1)^{pq} \mu \wedge \eta\,,

then follows. Therefore \wedge\, as a blinear map \wedge:\eta^p(M)\otimes \eta^q(M) \to \eta^{p+q}(M)\,. A less abstract representation of the exterior product can be obtained from the tensor product, as

\eta \wedge \mu = \eta \otimes \mu - \mu \otimes \eta\,.

Components

Given \eta \in \eta^p(M)\, with components \eta = \tfrac{1}{p!}\eta_{ a_1 a_2\cdots a_p}dx^{ a_1}\wedge dx^{ a_2}\wedge\cdots\wedge dx^{ a_p} and \mu \in \eta^q(M)\, with components \mu= \tfrac{1}{q!} \mu_{ b_1 b_2\cdots b_p}dx^{ b_1}\wedge dx^{ b_2}\wedge\cdots\wedge dx^{ b_q}, then

(\eta\wedge\mu)_{a_1 a_2 \cdots a_p\, b_1 b_2 \cdots b_q} = \frac{(p+q)!}{p! q!} \eta_{ \,[ a_1 a_2\cdots a_p} \mu_{ b_1 b_2\cdots b_q ]}\,.
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