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Suppose $f$ smoothly maps a domain ##U## of ##\mathbb{R}^2## into ##\mathbb{R}^3## by the formula ##f(x,y) = (x,y,F(x,y))##. We know that ##M = f(U)## is a smooth manifold if ##U## is open in ##\mathbb{R}^2##. Now I want to find the determinant of the metric in order to compute the area of the manifold

$$

I = \int 1 |g|^{1/2} d^2x

$$

I guess that the metric on ##\mathbb{R}^n## is the Kronecker delta, so that

$$

g_{ij} = \frac{d\xi^a}{dx^i} \frac{d\xi^b}{dx^j} \delta_{ab}

$$

So if I can find ##\xi^a##, my task is easy. How do I determine ##\xi^a##. Any hints/help/solutions? Thanks

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# A Find the determinant of the metric on some graph

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