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Calculus and Beyond Homework Help
Integral of a differential form
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[QUOTE="kiuhnm, post: 6119513, member: 578547"] [h2]Homework Statement [/h2] Suppose that a [I]smooth[/I] differential ##n-1##-form ##\omega## on ##\mathbb{R}^n## is ##0## outside of a ball of radius ##R##. Show that $$ \int_{\mathbb{R}^n} d\omega = 0. $$ [h2]Homework Equations[/h2] [/B] $$\oint_{\partial K} \omega = \int_K d\omega$$ [h2]The Attempt at a Solution[/h2] If ##\omega## is ##0## outside of the ball, by continuity, it must be ##0## on the surface of the ball as well. We know that $$ \omega = \sum_{i=1}^n f^i(x^1,\ldots,x^n) dx^1 \wedge \cdots \wedge \widehat{dx^i} \wedge \cdots \wedge dx^n $$ for some functions ##f^i:\mathbb{R}^n\to\mathbb{R}##, so $$ d\omega = \sum_{i=1}^n (-1)^{i-1} \frac{\partial f^i}{\partial x^i} dx^1\wedge\cdots\wedge dx^n. $$ All those partial derivatives are ##0## outside of the ball, so $$ \int_{\mathbb{R}^n} d\omega = \int_B d\omega = \oint_{\partial B} \omega = 0, $$ where ##B## is the ball. [/QUOTE]
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Integral of a differential form
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