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We say a weak derivative of $f$ satisfies $$\int \frac{df}{dx_i}g\;dx = -\int f \frac{dg}{dx_i}\;dx$$ for all $g \in C_c^\infty$.

I know the boundary disappears since $g$ is zero on the boundary. My question is is about the integration wrt $x_i$ and the $dx$ in the integral. Shouldn't the integral on the RHS have $dx_1...dx_{i-1}dx_{i+1}...dx_n$ instead of $dx$ (includes all the coordinates) since we integrated over $x_i?$ Or is that not what's going on? Please help.

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Actually we integrate over some domain $\Omega\subset \mathbb{R}^n$, where $f$ is defined. –  nikita2 Oct 14 '12 at 16:52

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No. Think of the one-dimensional case.

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