# Ngô Quốc Anh

## April 16, 2020

### Restriction of gradient, Laplacian, etc on level sets and applications

Filed under: Uncategorized — Tags: — Ngô Quốc Anh @ 18:10

This topic is devoted to proofs of several interesting identities involving derivatives on level sets. First, we start with the case of gradient. We shall prove

The first identity

$\displaystyle \partial_\nu f = \pm |\nabla f|$

on the level set

$\displaystyle \big\{ x \in \mathbf R^n : f(x) =0 \big\}.$

The above identity shows that while the right hand side involves the value of $f$ in a neighborhood, however, the left hand side indicates that only the normal direction is affected. Heuristically, any change of $f$ along the level set does not contribute to any derivative of $f$, namely, on the boundary of the level set, the norm of $\nabla f$ is actually the normal derivative $\partial_\nu f$. Therefore, the only direction taking into derivatives of $f$ is in the normal direction and this should be true for higher-order derivatives of $f$.

Next we prove the following

The second identity

$\displaystyle \partial_\nu \big(x \cdot \nabla f \big)=(\partial_\nu^2 f) (x \cdot \nu)$

on the level set

$\displaystyle \big\{ x \in \mathbf R^n : \partial_1 f(x) = \cdots = \partial_n f(x) = 0 \big\}.$

Combining the above two identities, we can prove

The third identity

$\displaystyle \partial_\nu^2 f=\Delta f$

on the level set

$\displaystyle \big\{ x \in \mathbf R^n : \partial_1 f(x) = \cdots = \partial_n f(x) = 0 \big\}$

which basically tells us how to compute the restriction of Laplacian on level sets. This note is devoted to a rigorous proof of the above facts together with a simple application of all these identities.