Let $$(M,g)$$ be an oriented Remannian surface. Then globally $$(M,g)$$ has a canonical area-2 form $$\mathrm{d}M$$ defined by $$\mathrm{d}M=\sqrt{|g|} \mathrm{d}u^1 \wedge \mathrm{d}u^2$$ with respect to a positively oriented chart $$(u_{\alpha}, M_{\alpha})$$ where $$|g|=\mathrm{det}(g_{ij})$$ is the determinant of the Remannian metric in the coordinate frame for $$u_{\alpha}$$.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# The area form of a Riemannian surface

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