The divergence theorem connects the flux (through surface) and divergence (in a volume) for any vector field.
This theorem expresses continuity. It isn't clear (to me) whether there is a conserved quantity associated with the continuity equation. It appears that this theorem would be equivalent to mass conservation, if the flux represented (say) fluid flow. In the general case of a vector field, I'm not sure what (if anything) is conserved.
I would like to know if this continuity implies a conserved quantity and an underlying symmetry, by the converse of Noether's theorem. If this (existence of symmetry/conserved quantity) is true for some vector fields but not all, what causes the distinction? Examples would be greatly appreciated.
While I don't have a strong background on Lagrangian mechanics, I'm happy to be directed to background reading that would help.
Best Answer
The integral theorem of Gauss, $$\int\limits_V \! d^3x \; \vec{\nabla} \cdot \vec{A}(\vec{x}) =\int\limits_{\partial V}\! d \vec{\sigma} \cdot \vec{A}(\vec{x}), \tag{1} \label{1}$$ is a purely mathematical statement. Taken by itself, it does not express "continuity" in any sense.
The concept of continuity (in the physical sense) comes into play once you have a scalar field (scalar density) $\rho(t, \vec{x})$ and a vector field (current density) $\vec{j}(t, \vec{x})$ related by the continuity equation $$\frac{\partial \rho(t,\vec{x})}{\partial t} + \vec{\nabla} \cdot \vec{j}(t,\vec{x}) =0. \tag{2} \label{2}$$ Defining the "charge" contained in a volume $V \subset \mathbb{R}^3$ at time $t$ by $$Q_V(t):= \int\limits_V \! d^3x \, \rho(t,\vec{x}), \tag{3} \label{3}$$ the integral theorem of Gauss \eqref{1} can be used to show that \eqref{2} implies $$\frac{d Q_V(t)}{dt}=\int\limits_V \! d^3x \, \frac{\partial \rho(t,\vec{x})}{\partial t}=-\int\limits_V\! d^3 x \, \vec{\nabla} \cdot \vec{j}(t,\vec{x})=-\int\limits_{\partial V} \! d \vec{\sigma} \cdot \vec{j}(t,\vec{x})=: -I_{\partial V}(t), \tag{4} \label{4} $$ relating the change of the charge contained in the volume $V$ to the flux (current) $I_{\partial V}(t)$ through the surface $\partial V$ of the volume $V$. Conversely, if $$\dot{Q}_V(t)=-I_{\partial V}(t)\tag{5} \label{5}$$ holds for "any" three-dimensional manifold $V \subset \mathbb{R}^3$ (subject to some mathematical qualification), the continuity equation \eqref{2} can be derived as the "local" version of \eqref{5}.
Assuming further that $\rho(t, \vec{x})$ and $\vec{j}(t,\vec{x})$ fall off sufficiently fast for $|\vec{x}|\to \infty$, \eqref{4} implies that the total charge $$Q:= \int\limits_{\mathbb{R}^3} \! d^3x \, \rho(t,\vec{x}) \tag{6} \label{6}$$ is time-independent, defining a conserved quantity.
Prominent examples are the charge density $\rho$ with the current density $\vec{j}$ in electrodynamics, the energy density of the electromagnetic field $\eta$ together with the energy flux density $\vec{S}$ in Maxwell's theory (in the absence of charges), mass density $\rho$ together with $\rho \vec{v}$ in nonrelativistic continuum mechanics and many others.