Past exam of the Probability, statistics and options II course of the University of Oxford / 2008 / F / 2 / b / Solution

Define $F(x)=x_1^2/a^2+x_2^2/b^2+x_3^2/c^2-1$. Its gradient is $2(x_1/a^2,x_2/b^2,x_3/c^2)$. It cannot vanish on $F^{-1}(0)$, because the only possible zero is the origin and that point has $F=-1$. The regular level-set theorem therefore makes the ellipsoid a two-dimensional embedded submanifold. Its normal space isand the tangent space is the orthogonal complement of that line.