The volume form on the unit sphere ##S^{n}## in ##\mathbb{R}^{n+1}## is given by(adsbygoogle = window.adsbygoogle || []).push({});

$$i_{{\bf r}}\ dx^{1}\wedge \dots \wedge dx^{n+1}=\sum (-1)^{i-1}x^{i}dx^{1}\wedge\dots \widehat{dx^{i}} \dots \wedge dx^{n+1}.$$

Why must the volume form ##dx^{1}\wedge \dots \wedge dx^{n+1}## act on the vector ##{\bf r}## to give the volume form on the unit sphere?

Also, how do I get the form of the volume form on the right-hand side of the equation?

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# A Volume form on unit sphere

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