\section{Mass Conservation \hfill 14.12.2023}
Let
\eqa
\Phi_t:&& \Omega_0\to \Omega_t,\quad \bx_0\mapsto\bx_{\ }=\bp(t,t_0,\bx_0),
\nonumber\\
\Phi_{-t}:&&\Omega_t\to \Omega_0,\quad \bx_{\ }\mapsto\bx_0=\bq(t,t_0,\bx)=\bp(t_0,t,\bx),
\nonumber
\aqe
be the flow of the vector field $\bV$ with single field lines
characterized by $\bp(t,t_0,\bx_0)$
\eqa
\pdtwo{\bp}{t}&=& \bV(t,\bp(t))\ =\ \bV\circ\bp,
\qquad\bp(t_0,t_0,\bx_0)\ =\ \bx_0,
\aqe
\i.e., for each starting point $\bx_0$ is $\bp$ the solution of
\eqa
\odtwo{\bx}{t}&=&\bV(t,\bx).
\aqe
$\Omega_t$ a volume comoving with the flow and $J$ the Jacobian
of the transformation,
\eqa
\Omega_t&=&\Phi_t(\Omega_0),
\quad J(t,t_0,\bx_0)\ =\ \pdtwo{\lp p^1,p^2,\cdots,p^n\rp}{
 \lp x_0^1,x_0^2,\cdots,x_0^n\rp}
\ =\ \det\lp\pdtwo{p^i}{x_0^k}\rp.
\aqe
\begin{figure}[hbtp]
\centering{
\includegraphics[width=0.55\textwidth]{figs/vlasov_1}
}
    \caption{
       Integral lines of a vector field.
          }
\end{figure}

The flow $x^i=p^i(t,t_0,\bx_0)$ induces a variable transformation
in the differential volume element (hodge dual of 1),
\eqa
*1&=&\frac{1}{n!}\rd x^{i_1}\wedge\rd x^{i_2}\wedge\cdots\wedge\rd x^{i_n}
\,\veps_{i_1i_2\ldots i_n}\sqrt{g(\bx)}
\nonumber\\ &=&
\frac{1}{n!}\lp\pdtwo{p^{i_1}}{x_0^{j_1}}\rd x_0^{j_1}\rp \wedge
\lp\pdtwo{p^{i_2}}{x_0^{j_2}}\rd x_0^{j_2}\rp \wedge\cdots\wedge
\lp\pdtwo{p^{i_n}}{x_0^{j_n}}\rd x_0^{j_n}\rp
\,\veps_{i_1i_2\ldots i_n}\sqrt{g\circ\bp}
\nonumber\\
&=&
\frac{1}{n!}\pdtwo{p^{i_1}}{x_0^{j_1}}
\pdtwo{p^{i_2}}{x_0^{j_2}}\cdots\pdtwo{p^{i_n}}{x_0^{j_n}}\,
\veps_{i_1i_2\ldots i_n}
\rd x_0^{j_1}\wedge\rd x_0^{j_2}\wedge\cdots\wedge\rd x_0^{j_n}
\,\sqrt{g\circ\bp}
\nonumber\\
&=&
\frac{1}{n!}J(t,t_0,\bx_0)\,\veps_{j_1j_2\ldots j_n}\sqrt{g\circ\bp(t,t_0,\bx_0)}\ 
\rd x_0^{j_1}\wedge\rd x_0^{j_2}\wedge \cdots\wedge\rd x_0^{j_n},
\aqe
where 
\eqa
\det(A)&=&\veps_{i_1i_2\cdots\i_n}a_{1i_1}a_{2i_2}\cdots a_{ni_n}
\nonumber
\aqe
has been used.

\eqa
&&\odtwo{}{t}\int_{\Omega_t}\rd x^1\wedge\rd x^2\wedge\cdots
\wedge \rd x^n\,\sqrt{g(\bx)}\,\rho(t,\bx)
\nonumber\\ 
&=&\odone{t}\int_{\Omega_0}\rd x_0^1\wedge\rd x_0^2\wedge\cdots\wedge \rd x_0^n\,J\lp\sqrt{g}\rho\rp\circ\bp
\nonumber\\ 
&=&\int_{\Omega_0}\rd x_0^1\wedge\rd x_0^2\wedge\cdots\wedge \rd x_0^n\,\pdone{t}\lsp J\lp\rho\sqrt{g}\rp\circ\bp\rsp
\nonumber\\
&=&\int_{\Omega_0}\rd x_0^1\wedge\rd x_0^2\wedge\cdots\wedge \rd x_0^n\,\lsp
\pdone{t}\lp{\sqrt{g\circ\bp}\,J}\rp\,\rho\circ\bp
+J\sqrt{g\circ\bp}\,\lp\pdtwo{\rho}{t}
+V^k\pdtwo{\rho}{x^k}\rp\circ\bp\rsp
%\nonumber\\
%&=&\int_{\Omega_0}\rd x_0^1\wedge\rd x_0^2\wedge\cdots\wedge \rd x_0^n\,J\lsp\sqrt{g}\rho\pdtwo{V^k}{x^k}+
%\lp\sqrt{g}\pdtwo{\rho}{t}+V^k\pdtwo{\sqrt{g}\rho}{x^k}\rp\rsp\circ\bp
\nonumber\\
&=&\int_{\Omega_0}\rd x_0^1\wedge\rd x_0^2\wedge\cdots\wedge \rd x_0^n\,
J\sqrt{g\circ\bp}
\lp\rho\frac{1}{\sqrt{g}}\pdone{x^k}\lp\sqrt{g}V^k\rp
+\pdtwo{\rho}{t}+V^k\pdtwo{\rho}{x^k}
\rp\circ\bp
\nonumber\\
&=&\int_{\Omega_t}\rd x^1\wedge\rd x^2\wedge\cdots\wedge \rd x^n\,
\sqrt{g}\,
\lsp\rho\,\Div\bV+\pdtwo{\rho}{t}+\bV\cdot\nabla\rho\rsp
\nonumber\\
&=&\int_{\Omega_t}\rd x^1\wedge\rd x^2\wedge\cdots\wedge \rd x^n\,
\sqrt{g}\,
\lsp\pdtwo{\rho}{t}+\Div\lp\rho\bV\rp\rsp.
\aqe

Time derivative of Jacobian,
\eqa
\bA^{-1}\bA&=&\bI,\quad \bC^T\bA\ =\ \det(\bA)\bI,\quad
J\,\delta^i_{\ k}\ =\ c_j^{\ i}a^j_{\ k},
\nonumber\\
\pdone{t}J&=&\pdone{t}\lp c_j^{\ (i)}a^j_{\ (i)}\rp
\ =\ c_j^{\ i}\pdone{t}\lp a^j_{\ i}\rp
\ =\ c_j^{\ i}\pdone{t}\lp\pdtwo{p^j}{x_0^i}\rp
\nonumber\\
&=&c_j^{\ i}\pdone{x_0^i}\pdtwo{p^j}{t}\
\ =\ c_j^{\ i}\pdone{x_0^i}\lp V^j\circ\bp\rp
\ =\ c_j^{\ i}\lp\pdtwo{V^j}{x^k}\circ\bp\rp\pdtwo{p^k}{x_0^i}
\nonumber\\
&=&c_j^{\ i}a^k_{\ i}\pdtwo{V^j}{x^k}\circ\bp
\ =\ J\,\delta_j^k\pdtwo{V^j}{x^k}\circ\bp \ =\ J\pdtwo{V^i}{x^i}\circ\bp.
\\
\pdone{t}\lp\sqrt{g\circ\bp}\rp&=&\lp V^k\pdone{x^k}\sqrt{g}\rp\circ\bp
\\
\pdone{t}\lp\sqrt{g\circ\bp}\,J\rp&=&\sqrt{g\circ\bp}\,J
\lsp\frac{1}{\sqrt{g}}\pdone{x^k}\lp\sqrt{g}V^k\rp\rsp
\ =\ J\lp\sqrt{g}\,\Div\bV\rp\circ\bp.
\aqe
\clearpage

Change of differential volume valid for the flow of any time dependent
vector field

\feqa
\pdone{t}\lp\sqrt{g\circ\bp}\,J\rp&=&J\lp\sqrt{g}\,\Div\bV\rp\circ\bp.
\aqef

Conservation of total mass implies

\feqa
\pdtwo{\rho}{t}+\Div\lp\rho\bV\rp&=&0.
\aqef

Liouville's theorem says that in case of zero divergence of the vector field
$\bV$, the mass density along field lines is constant,

\feqa
\odone{t}\lp\rho\circ\bp\rp\ =\ \pdtwo{\rho}{t}+\bV\cdot\nabla\rho&=&0.
\aqef

The theorem also says that phase space volume is
conserved even in the case of a time dependent vector field.

In kinetic plasma theory, particle conservation governed by the
single particle distribution function
\eqa
\int\rd x^3\rd v^3 f(t,\bx,\bv)&=&N,
\aqe
and the divergence free Hamiltonian vector field Vlasov*s equation
is obtained 

\feqa
\pdtwo{f}{t}+\bv\cdot\pdtwo{f}{\bx}+\frac{e}{m}
\lp\bE+\frac{1}{c}\bv\times\bB\rp\cdot\pdtwo{f}{\bv}&=&0.
\aqef

Expressed by Liouville's theorem, the distribution function is constant
along the integral lines of Hamiltonian vector fields.

The characteristic equations are
\eqa
\dot\bx&=&\bv,\nonumber\\
\dot\bv&=&\frac{e}{m}\lp\bE+\frac{1}{c}\bv\times\bB\rp,\nonumber
\aqe
and, therefore,
\eqa
\Div\bV&=&\pdone{\bx}\cdot\bv+\pdone{\bv}\cdot
\frac{e}{m}\lp\bE+\frac{1}{c}\bv\times\bB\rp\ =\ 0,\nonumber
\aqe
because 
\eqa
\pdone{\bv}\cdot\lp\bv\times\bB\rp&=&0.\nonumber
\aqe
