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
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
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
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.