# Parametrized Surface Problem

1. Nov 9, 2009

### Sistine

1. The problem statement, all variables and given/known data
If a regular surface can be parametrized in the form

$$s(u, v) = a(u) + b(v)$$

where a and b are regular parametrized curves with $$(u,v)$$ in some domain $$D\subseteq\mathbb{R}^2$$, show
that the tangent planes along a fixed coordinate curve of $$D$$ are all parallel to a line.

2. Relevant equations
The normal to the tangent plane at $$(u_0,v_0)$$ is given by

$$N(u_0,v_0)=\partial_1s(u_0,v_0)\times\partial_2s(u_0,v_0)$$

where

$$\partial_is=\frac{\partial s}{\partial x_i}$$

3. The attempt at a solution
Taking my fixed coordinate curve in $$D$$ to be

$$L:=\{(u_0,t) | t\in\mathbb{R}\}$$

I found the normal using the equation above to be

$$N(t)=a'(u_0)\times b'(t)$$

But a basis for the $$b'(t)$$'s is at most 3 dimensional, for $$N(t)$$ to be orthogonal to a line (and thus the tangent plane to be parallel to a line) a basis for $$N(t)$$ would have to be 2 dimensional. So I cannot show the statement of the problem. Am I doing something wrong, is there a better way to approach the problem?