Motivated by the  classical  formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$  and  $L_{X} \Omega=Div(X)  \Omega$  and  the  essential role of the diff operator $d$ in definition of divergence, we define  some  quantities, say $\overline{Div}$, based on the  adjoint operator $d^{*}=\pm *d*$, where $*$  is  the  Hodge star operator.

 In this way our  main question is that:

>What are  some   geometric or physical interpretations  for $\overline{Div}$? What are some calculus  identities for this  quantity?In particular is it true  that for  a  closed  manifold $M$, with volum  form  $\Omega$, we  have $\int_{M} \overline{Div}(X)\Omega=0$?

1. For a vector  field  $X$ on  a   $2$  dimensional  surface with  volum form $\Omega$ define:

$$\overline{Div}(X)=(i_{X}\circ d^{*}+d^{*}\circ i_{X})(\Omega)$$

2. A  vector  field  $X$ on a Riemannian manifold $(M,g)$  defines a one  form $\alpha$.  Now   $\overline{Div}(X)$  is  defined as a  unique  function with  $$\alpha \wedge d^{*}(\Omega)=\overline{Div}(X). \Omega    $$


3.  For  a  symplectic  manifold  $(M,\omega)$,  $\overline{Div}$  is  the  unique  function with  $$  (i_{X}\circ d^{*}+d^{*}\circ i_{X})(\Omega)\wedge \omega=\overline{Div}(X).\Omega$$    where  $\Omega$ is  the  corresponding  volume form.

4. What is the dynamical interpretation of  $\overline{Div}(X)=0$. This is motivated by classical case: If $Div(X)=0$ then $X$  has  no  an attractor since the flow generates a one parameter family of  volume preserving  diffeomorphisms.