Prove that if (mathcal{S}) is the part of a graph (z=g(x, y)) lying over a domain (mathcal{D})

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Prove that if \(\mathcal{S}\) is the part of a graph \(z=g(x, y)\) lying over a domain \(\mathcal{D}\) in the \(x y\)-plane, then

\[
\iint_{\mathcal{S}} \mathbf{F} \cdot d \mathbf{S}=\iint_{\mathcal{D}}\left(-F_{1} \frac{\partial g}{\partial x}-F_{2} \frac{\partial g}{\partial y}+F_{3}ight) d x d y
\]

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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