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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.

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    $\begingroup$ In case you might need some references, see Gerald B, Folland's Advanced Calculus. Sec. 5.5 (on the divergence thrm.) talks about the concern's in Roland F.'s post, while Sec, 5.6 (applications to mathematical physics) derives the continuity equation via the method's used by Hyperon. $\endgroup$ Feb 9 at 14:21
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    $\begingroup$ The above references do not discuss any connection to Noether's thrm. The applicability of the divergence thrm. is independent of any underlying symmetries of the vector field. $\endgroup$ Feb 9 at 14:31
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    $\begingroup$ You are sorely abusing the notions of continuity here. Very briefly, mathematicians invented continuity for the purposes of doing calculus. Physical phenomena such as gas laws were found experimentally to abide by calculus formula, but there wasn't anything in the atomic models of the time to justify the math's continuity requirement. So physicists formalized continuity into their atomic physical models as a backstop decades after the use of calculus had become routine. Continuity was just an overdue patch. $\endgroup$
    – Phil Sweet
    Feb 10 at 2:19
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    $\begingroup$ vaguely related and possibly helpful - engineering.stackexchange.com/q/51881 $\endgroup$
    – Phil Sweet
    Feb 10 at 2:25
  • $\begingroup$ Thanks for the corrections and references @AlbertusMagnus, Phil Sweet $\endgroup$ Feb 11 at 5:40

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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.

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  • $\begingroup$ This cleared a lot of my misconceptions, and is exceptionally well written. $\endgroup$ Feb 11 at 5:41
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The coordinate independent form for the divergence of vector field in the tangent space of a metric manifold with metric matrix $g_{ik}$ is $$\frac{1}{\sqrt{\det g}}\ \sum_{ik}\partial_{x_i}\ \left( \sqrt{\det g} g^{-1}_{ik}\quad v_k(x) \right ). $$

So the use of the divergence formula is assuming the existence of a scalar product on gradients and general vector fields. In good old books the divergence is defined primarly as the volume limit of the radial outward flow over the surface of a small euclidean sphere $S_{2(x,r)}$ of radius r

$$\text{div} v(x) = \lim_{r\to 0} \frac{1}{V(r)} \ \int_{\partial_{S_2(x,r)}} v\cdot dA $$

Its evident, that the $\text{div}$ is tied to conserved flow or local production volume densitiy in an euclidean geometry.

This definition is crucial for distributions the vacuum solution of the Coulomb field of a point source at the orogin

$$\nabla \cdot \frac{ \vec x} {|x|^{3/2}} = 0 $$

where the definition by derivatives is not applicable.

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    $\begingroup$ For example, in Feynmann's Lectures on Physics Vol. II, he derives the divergence of a vector field in the form given in your post. It does indeed express a concept of what "what flows in must flow out", however, it is not so strong as to express "local conservation", i.e. that the vector field may have "sources" or "sinks" which would violate the continuity equation for some specific volume. $\endgroup$ Feb 9 at 14:13

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