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?