Let $D \subset \mathbb{R}^n$ be bounded and $f$ a smooth compactly supported function such that its support is contained within $D$. I am interested in$$\int_D \nabla f dx.$$If $n = 1$, then by the Fundamental Theorem of Calculus it is obvious that this integral is zero. However I am less sure when $n > 1$. Is the integral still equal to zero? If so, how can one see this?
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