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Compute the differentials of the following differential forms.

$\text{(a)}\,\,\,\alpha=\sum_{i=1}^n(-1)^{i-1}\,x_i\,\mathrm dx_1\wedge\cdots\wedge\mathrm dx_{i-1}\wedge\mathrm dx_{i+1}\wedge\cdots\wedge\mathrm dx_n.$

$\text{(b)}\,\,\,r^{-n}\alpha,\text{ where $r=[x_1^2+\cdots+x_n^2]^{1/2}$}$.

$\text{(c)}\,\,\,\sum_{i=1}^ny_i \,\mathrm d x_i,\text{ where $(x_1,\ldots,x_n,y_1,\ldots,y_n)$ are coordinates in $\mathscr R^{2n}$}$.

I did the part a, the answer is the n. For part b, I tried to get r^(-n)*a, and I tried to use induction for this part. I tried n=1, n=2, but I get nothing. I think I may need a new way to solve this.

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  • $\begingroup$ @Adobe Please be more careful in the edits - there is a missing $y_i$ in $(c)$ $\endgroup$ – L. F. Dec 20 '13 at 22:03
  • $\begingroup$ @L.F. Thanks! And sorry! $\endgroup$ – user93957 Dec 20 '13 at 22:05
  • $\begingroup$ @L.F. I can't see the original version of the question but I think it had $x_i$ in (a), not $x_1$. $\endgroup$ – Andrey Sokolov Dec 24 '13 at 2:52
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I will assume you are asked to compute the exterior derivatives of these differential forms. Since $\alpha$ and $r^{-n}\alpha$ are the $(n-1)$-forms, their exterior derivatives will be $n$-forms. To find $d\alpha$ you just need to apply the definition of the exterior derivative (in local coordinates) and the properties of the exterior product $\wedge$ such as associativity and anticommutativity. For (b), you can do the same as in (a) or you can make use of $$d(r^{-n}\alpha)=d(r^{-n})\wedge\alpha+r^{-n}d\alpha,$$ where $d\alpha$ is the $n$-form you found in (a) and $d(r^{-n})$ is a 1-form you get by taking the exterior derivative of $r^{-n}$.

This wikipedia page might be useful http://en.wikipedia.org/wiki/Exterior_derivative

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