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Let $A$ be open in $R^k$, let $\alpha : A \rightarrow R^n$ be of class $C^\infty$. Let $x$ denote the general point of $R^k$, let $y$ denote the general point of $R^n$ If $I = (i_i, ...,i_l)$ is an ascending l-tuple from the set $\{1,...,n\}$ then

$\alpha^*(dy_i) = \sum_{[J]} (\det\frac{\partial\alpha_I}{\partial x_J})dx_J$ Here $J = (j_1,...,j_l)$ is an ascending l-tuple from the set $\{1,...,k\}$ and $\frac{\partial\alpha_I}{\partial x_J}$= $\frac{\partial(\alpha_{i_1},...,\alpha_{i_l})}{\partial(x_{j_1},...,x_{j_l})}$

Attempt at the solution :

We know $\alpha^*(dy_i) =d\alpha_i$, then we can write $\alpha^*(dy_{i_1}) \wedge \cdots \wedge \alpha^*(dy_{i_l})$ = $d\alpha_{i_1} \wedge \cdots \wedge d\alpha_{i_l}$ = $\sum_{[J]} \frac{\partial \alpha_{i_1}}{\partial x_J}dx_J$ $\wedge \cdots \wedge \sum_{[J]} \frac{\partial \alpha_{i_l}}{\partial x_J}dx_J$ = $ \sum_{[J]} (\det\frac{\partial\alpha_I}{\partial x_J})dx_J$ as required. I'm not too confident about my solution, in particular I get really confused about the indices, if someone could point out my mistakes and provide any additional insight I would be really grateful. Thank you in advance

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