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I think that the Wikipedia article Surface Area says that a function $Area(S)$ defined on piecewise smooth surfaces which (i) agrees with the area of common surfaces, (ii) is additive over surfaces, and (iii) is invariant under Euclidean motions, is given by the usual integral $\iint_A |\mathbf{r}_u \times \mathbf{r}_v|\, dA$.

I'd like a citation to this result.

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