For a given smooth function on manifold , the gradient of is given by

.

Note that gradient of is also a vector field on . Thus, for each , it is reasonable to talk about .

Definition 1. Hessian of , denoted by , is defined as the symmetric -tensor.

We also denote by the following

.

Thus

Note that

.

Since then

which implies

.

Definition 2. Laplacian of , denoted by , is defined as the trace of .

Note that is a -tensor, then in local coordinates, one has

.

It is clear that is a -tensor field. To see this fact, one can assume then from

one has

since which is exactly an -tensor. Then we can define divergence of a vector field as following

Definition 3. Divergence of vector field is given by.

In coordinates, this is

and with respect to an orthornormal basis

.

Thus .

**NOTICE**: To avoid any inconvenience caused, from now we denote gradient of by instead of . This is because is covariant derivative of , this is an -tensor instead of a vector field as mentioned in this entry.

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