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Don't know how the prove this apparently simple relation: if $x(u,v)$ is a chart of a surface $S$, with unit normal $N(u,v)$, then $N_u\times N_v=Kx_u\times x_v$, where $K$ is the Gaussian curvature. I can write $$N=\dfrac{x_u\times x_v}{\lvert\lvert x_u\times x_v\rvert\rvert}$$ and compute all the derivatives, but this does not seem nice. I think it should be really simple, I am just not seeing the point.

Thank you!

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You might start by recalling (1) the definition of $K$, and (2) the matrix of what some people call the shape operator (the derivative of the normal map $N$) with respect to the basis $\{x_u,x_v\}$.

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