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 $\bar{Div}$, based on the adjoint operator $d^{*}=\pm *d*$.
In this way our main question is that:
What are some geometric or physical interpretations for $\bar{Div}$? 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 $\int_{M} \bar{Div}(X)\Omega=0$?
- For a vector field $X$ on a $2$ dimensional surface with volum form $\Omega$ define:
$$\bar{Div}(X)=(i_{X}\circ d^{*}+d^{*}\circ i_{X})(\Omega)$$
A vector field $X$ on a Riemannian manifold $(M,g)$ defines a one form $\alpha$. Now $\bar{Div}(X)$ is defined as a unique function with $$\alpha \wedge d^{*}(\Omega)=\bar{Div}(X). \Omega $$
For a symplectic manifold $(M,\omega)$, $\bar{Div}$ is the unique function with $$ (i_{X}\circ d^{*}+d^{*}\circ i_{X})(\Omega)\wedge \omega=\bar{Div}(X).\Omega$$
where $\Omega$ is the corresponding volum form.