I have a smooth diffeomorphism $\Phi\colon M \to N$ between two 2D hypersurfaces in $\mathbb{R}^3$, eg. the sphere and some deformed version of the sphere.
If I have a unit (outward) normal vector $n$ at some point $p \in M$, is there a formula to get the unit (outward) normal $\tilde n$ at the point $\Phi(p) \in N$ in terms of $n$?
Maybe (or not) something like $\tilde n = M\Phi(n)$ (where $\Phi$ is applied to each component on $n$, and $M$ is some matrix)?
I need this because I wish to describe the normal field of $N$, which is a complicated surface, using the normal field on $M$ (which is the sphere in my example).