Definition of second fundamental form

Question

$F:R^n\supset U \rightarrow F(U)\subset M\subset R^{n+1}$ is a local represent of Riemannian manifold $M$. $\nu$ is outward normal vector.How to show $$ (\partial_{x_i}\nu,\partial_{x_j}F)=(\nu,\partial_{x_i}\partial_{x_j}F) $$

Answer 1

$(\partial_{x_i}\nu,\partial_{x_j}F)=\partial_{x_i}\underbrace{(\nu,\partial_{x_j}F)}_{=0}-(\nu,\partial_{x_i}\partial{x_j}F)$

the product-rule and the first product is zero because the outward normal vector is orthogonal to the tangent vectors.