  Two kinds of differential operators that can be generally defined on an
arbitrary smooth surface in a finite dimensional Euclid space are studied, one
is termed as surface gradient and the other one as Levi-Civita gradient. The
surface gradient operator is originated from the differentiability of a tensor
field defined on the surface. Some integral and differential identities have
been theoretically studied that play the important role in the studies on
continuous mediums whose geometrical configurations can be taken as surfaces
and on interactions between fluids and deformable boundaries. The definition of
Levi-Civita gradient is based on Levi-Civita connections generally defined on
Riemann manifolds. It can be used to set up some differential identities in the
intrinsic/coordiantes-independent form that play the essential role in the
theory of vorticity dynamics for two dimensional flows on general fixed smooth
surfaces.
