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Problem If a coordinate neighborhood of a regular surface can be parametrized as follows:

*x*$(u,v)$ = $\alpha_{1}$$(u)$ + $\alpha_{2}$$(v)$

where $\alpha_{1}$ and $\alpha_{2}$ are parametrized regular curves, show that the tangent planes along a coordinate fixed curve in this neighborhood are all parallel to a single line.


Well, we have that x{u} = $\alpha_{1}^{'}$$(u)$ and x{v} = $\alpha_{2}^{'}$$(v)$, which is a basis for the tangent plane at some point. I couldn't go further. Could you help me?

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Fix $u \equiv u_0$. At a point $(u_0, v)$, the tangent plane to the surface at $x(u_0, v)$ is spanned by the vectors $\alpha_1'(u_0)$ and $\alpha_2'(v)$. This means that any line with a direction vector $\alpha_1'(u_0)$ will be parallel to all tangent planes along $u \equiv u_0$.

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