Let the be the coordinates of and be the coordinates of . Denote . Suppose is an embedding map with where .

Then , form a basis for at every . Define the metric on as

is a function, we can define the tangential gradient of as

Here we view as a function on and

Consider the case when is a graph of some function.

Then , here .

and .

The unit normal vector of is .

Naturally, the tangential vector of should be defined as

here is respect to . Verify that this definition of tangential gradient is consistent with

Let us find the explicit expression of .

here

Next, let us calculate

,

Expand this by using the fact , one will see this is equal to

**But there should have a simple way to understand how is defined.** Actually, this can be seen from the identity

holds for every vector field on . The definition of should satisfies , which can also be seen from chern’s book.

Plugging in in , we get

considering , we get

Klaus Ecker, *Regularity theory for mean curvature flow*. Appendix A