Chapter 7: Problem 9
Show that \(\omega_{1} \wedge \cdots \wedge \omega_{k}\) is a \(k\)-form.
Chapter 7: Problem 9
Show that \(\omega_{1} \wedge \cdots \wedge \omega_{k}\) is a \(k\)-form.
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Show that every 2 -form in the three-dimensional space \(\left(x_{1}, x_{2}, x_{3}\right)\) is of the form $$ P x_{2} \wedge x_{3}+Q x_{3} \wedge x_{1}+R x_{1} \wedge x_{2} . $$
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Show that the boundary of the boundary of any chain is zero: \(\partial \partial c_{k}=0\).
Show that the product of monomials is associative: $$ \left(\omega^{k} \times \omega^{r}\right) \wedge \omega^{m}-\omega^{2} \times\left(\omega^{2} \times \omega \omega^{m}\right) $$ and skew-commutative: $$ \omega^{k} \wedge \omega^{l}=(-1)^{k} \omega^{l} \wedge \omega^{k} . $$
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