I am doing some computer science where we select actions from a set $X \subset \mathbb R^d$ whose boundary is a smooth manifold with some curvature conditions. The embedding in $ \mathbb R^d$ is important for my purposes. I am having great difficulty getting a reference for the following sort of facts:
Suppose $\{x \in U: F(x)=0\}$ is a parametrised coordinate patch. Let $N(x) = \frac{\nabla F(x)}{\|\nabla F(x)\|}$ be the unit normal funcction. The derivative of the normal unit vector map (in standard coordinates) is $M(x) = \frac{\nabla^2 F(x)}{\|\nabla F(x)\|} - g(x)\nabla F(x) $ for some scalar function $g(x)$.
Note: The above is a matrix representation of the second fundamental form.
For any $v$ tangent to the manifold at $x$ we have $M(x)v$ also tangent to the manifold.
For any $u,v$ tangent to the manifold at $x$ we have $w^T M(x) v = \frac{w^T\nabla^2 F(x)v}{\|\nabla F(x)\|} $.
What I can find is (a) loads of stuff about the sort of intrinsic geometry I learnt at school and (b) some more limited stuff about parametrised curves and surfaces. But I cannot find anything similar to the above. Does anyone know a good reference?