# Why can the gaussian curvature be computed this way?

I understood everything up to the last part ("It follows that the gaussian curvature $$K = K(s, v)$$ of the tube is given by..."). Why? What am I missing here? I know we can just compute it from the second fundamental form, but I want to understand what was done here.

If so, have a look at the discussion at page 166. There you see the equation $$dN(w_1) \wedge dN(w_2) = K w_1 \wedge w_2,$$ for a basis $$\{w_1, w_2\}$$ of a tangent space at a point. This is exactly what you need.