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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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I you write, for example 5 det Dg in TeX, it looks like this: $5 det Dg$, with no spacing between 5 and det or between det and Dg, and with "det" italicized. But if you write 5 \det Dg, with a backslash, then not only is "det" not italicized, but proper spacing precedes and follows it thus: $5\det Dg$. That is standard TeX usage. I edited the question accordingly. –  Michael Hardy Oct 28 '12 at 22:58
    
@MichaelHardy thanks I will keep that in mind for future. –  michael kar Oct 28 '12 at 22:59
    
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? –  anorton Oct 28 '12 at 23:37

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