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 identityon the level set

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

Next we prove the following

The second identityon the level set

Combining the above two identities, we can prove

The third identityon the level set

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.