N-form

$n-$forms

Definition. (Citation Needed) Let ${\omega }_1,{\omega }_2,\dots ,{\omega }_n$ be 1-forms defined on the Euclidean tangent space $T_p{\mathbb{R}}^n$ and $v_1,v_2,\dots ,v_n\in T_p{\mathbb{R}}^n$ be vectors. The $n-$form spanned by ${\omega }_1,\dots ,{\omega }_n$ is given by

\omega_1(v_2)&\omega_2(v_2)&\ldots&\omega_2(v_2)\\ \vdots&\vdots&&\vdots\\ \omega_1(v_n)&\omega_2(v_n)&\ldots&\omega_n(v_n)\\\end{vmatrix}$$ ## Interpretation In general, we define an $n-$form to be any alternating, multilinear real-valued function which acts on $n-$tuples of vectors. ## Reference [[Bachman, D. - A geometric approach to differential forms|@Bac12, Ch. 3]] [//begin]: # "Autogenerated link references for markdown compatibility" [1-form]: 1-form "1-form" [//end]: # "Autogenerated link references"