I am reading through a book which states that for an ellipse specified by the points that satisfy:
$f(x_2,x_3)=x_2^2/a^2 + x_3^2/b^2 = 1$
the unit normal is given by
$\mathbf{n}=\frac{\nabla f}{|\nabla f|}=\frac{2}{|\nabla f|} \left[\frac{x_2}{a^2}\mathbf{e_2} + \frac{x_3}{b^2}\mathbf{e_3}\right] = b^2 x_2\mathbf{e_2} + a^2x_3 \mathbf{e_3}$
where $\mathbf{e_i}$ denotes the unit vector in the $i^{\mathrm{th}}$ direction. This of course implies that $|\nabla f| = 2/a^2b^2$, but I can't seem to derive that magnitude of $|\nabla f|$ myself. Could someone help me out?
The unit normal is not correct. Take $a=b=2, x_2=0, x_3=2$, then $b^2x_2\textbf{e}_2+a^2x_3\textbf{e}_3=8\textbf{e}_3$, which is clearly not 0f unit magnitude.