I am currently self learning some differential forms through do Carmo's Differential Forms and Applications. In the first chapter, the author gives two related definitions of exterior product. I want to show that they are compatible with each other.

(1) Given 1-forms $\varphi_1,\dots, \varphi_k$, the exterior product is defined by $$\varphi_1\wedge\dots\wedge\varphi_k(\mathbf{v}_1,\dots,\mathbf{v}_k)=\det(\varphi_i(\mathbf{v}_j))$$ (2) Given a $k$-form $\omega$ and an $s$-form $\varphi$ in $\mathbb R^n$, with k-tuples $I=(i_1,\dots,i_k)$ and $J=(j_1,\dots,j_s)$ such that the $i$'s and $j$'s are strictly increasing and they take values from $\{1,\dots,n\}$. Then we can write $\omega=\sum_I a_I dx_I$ and $\varphi =\sum_J b_J dx_J$. And the exterior product is defined to be $$\omega \wedge \varphi=\sum_{IJ}a_Ib_Jdx_I\wedge dx_J$$ where the product on the right is from (1)

Using the definition of (2), if we have $1$-forms $\varphi_1,\dots, \varphi_k$, and given $\mathbf{v}_1,\dots,\mathbf{v}_n\in \mathbb R^n$, $J=\{1,\dots, n\}$, then we can write $\varphi_i=\sum_{J}a_{ij}dx_{j}$. We have $$\varphi_1\wedge\dots\wedge\varphi_k=\sum_{\underset{\text{k copies}}{\underbrace{J\times \dots\times J}}}a_{1j_1}\dots a_{kj_k}dx_{j_1}\wedge\dots\wedge dx_{j_k}$$ My attempt was to write each vectors into the linear combination of basis vectors and verify that both expressions agree for all combinations. However all these indices have confused me and I could not see a simple way to write it down.

Another approach is to use linearity and expand on definition (1). So for example, if we take $\varphi_i$ as a summation of $dx_j$'s, we get terms of the form $a_1dx_{i_1}\wedge \dots\wedge a_k dx_{i_k}(\mathbf{v}_1,\dots,\mathbf{v}_k)= \det(a_mv_{i,m})$, where $v_{i,m}$ denotes the $m^{\text{th}}$ component of the vector $\mathbf{v}_i$. But it is difficult to see how to group all terms together.

I would like to have hints on what approach could I take to solve this. And in case if my approach was correct, how does one write it cleanly with all these indices. Thanks in advance.

  • 1
    $\begingroup$ Here's a small suggestion: By multilinearity (of wedge and of determinant), it will suffice to check equivalence of the definitions when you take $\varphi_i$ to be basis $1$-forms. $\endgroup$ – Ted Shifrin Oct 5 '17 at 18:31
  • $\begingroup$ @TedShifrin Thanks Prof. Shifrin for the suggestion, now it's much simpler to work with. Also I'd like to express gratitude towards your lecture series on youtube. It has helped me a lot. $\endgroup$ – lEm Oct 6 '17 at 7:41

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