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I want to integrate $\int \nabla f(x)$ by substitution. Let $x = g(y)$.

Then is $\int \nabla f(x)$ equal to $\int \nabla [f(g(y))] |\det Dg| $ or $\int \nabla f|_{g(y)} |\det Dg| ?$

I am confused.

(PS: I know here I'm integrating a vector so think of it as $\nabla f \cdot \nabla f$ if you like.)

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    This question doesn't make much sense... $\nabla f(x_0, ..., x_i) = \frac{\partial f}{\partial x_0}e_0 + ... + \frac{\partial f}{\partial x_i}e_i$ where the $e$ are unit vectors in direction of the coordinate axes. A vector in one dimension isn't much use, right?2012-10-28

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