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Let $$u^{i}=\Phi^{i}(v^1,v^2)$$ be a change of variables (so $$\Phi: V \to U$$ is the diffeomorphism of the coordinate change). Calculate the effect on $$\sqrt{|g|}$$ and $$\mathrm{d}u^1 \wedge \mathrm{d}u^2$$ to prove $$\mathrm{d}M$$ is independent of the choice of positively oriented coordinates.