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For a slowly varying parallel shear flow $U(y){\mathbf{e}}_x$, the leading WKB method neglects mean-velocity gradients while retaining local advection. The continuity equation and linearized Euler equations then give the displayed acoustic dispersion relation. On the positive intrinsic-frequency branch, $\omega =U{k}_x+c_0|\mathbf{k}|$ and the group velocity is $U{\mathbf{e}}_x+c_0\mathbf{k}/|\mathbf{k}|$. A wavefront normal is parallel to $\mathbf{k}$; the acoustic energy-ray direction generally differs because of the advection term.

For a barotropic fluid with pressure $P=P(\rho )$, choose an energy density $U$ satisfying $\rho U^{\prime }(\rho )-U=P$. The continuity equation then gives ${\partial }_{t}U+\nabla \cdot (U\mathbf{u})=-P\nabla \cdot \mathbf{u}$. For an isothermal equation of state $P=c_s^2\rho$, one choice is $U=c_s^2\rho \ln (\rho /{\rho }_{\mathrm{ref}})$. A fixed reference density changes $U$ only by a conserved mass term. The same construction holds with surface density and vertically integrated pressure. This closure energy need not equal the microscopic internal energy of a gas maintained at fixed temperature.

Steady spherical continuity equation gives $r^2\rho u=\text{constant}$. Combining this with Euler equations for an inviscid fluid and $c_s^2=dp/d\rho$ gives the displayed equation. Smooth transonic passage requires $u^2=c_s^2$ and ${\Phi }^{\prime }=2c_s^2/r$ simultaneously. The barotropic enthalpy function gives the integral $u^2/2+h+\Phi =\text{constant}$.

Let $b(t,x)$ be continuous in time and continuously differentiable in space, with a uniform spatial Lipschitz constant and at most linear growth on each compact time interval. Its characteristic flow map exists forward and backward over every bounded time interval. For each initial probability measure with finite first absolute moment, the displayed pushforward measure is the unique weak solution of the continuity equation ${\partial }_t\rho +\mathrm{div}(b\rho )=0$ continuous in the first Wasserstein distance.

The Gronwall inequality bounds the flow by $C_T(1+|x|)$ and therefore preserves finite first absolute moments. The chain rule establishes the weak equation. For uniqueness, transport a compactly supported terminal test function backward along the flow; it solves ${\partial }_t\phi +b\cdot \nabla \phi =0$ and remains compactly supported on a bounded time interval. Inserting it into the weak identity identifies every terminal integral with that of the claimed pushforward measure. A zero divergence makes the corresponding scalar transport equation and the continuity equation coincide.

Charge conservation says that the total signed electric charge of an isolated collection stays constant. More generally, a collection closed to electric charge exchange has constant total electric charge, even if energy crosses its boundary. This is the charge bookkeeping relevant to a closed system or an isolated system; isolation from energy exchange is not required. Positive and negative electric charge count with opposite signs. Creating or removing an electron and a Positron together is compatible with charge conservation, because their combined electric charge is zero. ^[ref].

Imagine a fixed box initially containing electric charge $Q(0)=10\mu \mathrm{C}$. Stipulate incoming conventional current $I_{\mathrm{in}}=3\mu \mathrm{A}$ and outgoing conventional current $I_{\mathrm{out}}=1\mu \mathrm{A}$ throughout $0\le t\le 2\mathrm{s}$, with no other boundary transport. Since one ampere is one coulomb per second, the stored electric charge grows at $2\mu C/s$:If instead the outgoing electric current also equals $3\mu \mathrm{A}$, the stored electric charge remains $10\mu \mathrm{C}$ while electric charge passes through. These are stipulated transport examples; constant terminal electric currents alone do not guarantee a time-independent interior charge density.

Locally, charge conservation is the charge continuity equation, an instance of the general continuity equation:Here $\rho (\mathbf{r},t)$ is signed volume charge density and $\mathbf{J}(\mathbf{r},t)$ is current density. A positive divergence of current density means net outward transport from a small neighborhood, so its charge density decreases. For a fixed control volume $V$ with closed surface $\partial V$ and outward unit normal $\mathbf{n}$, the equivalent integrated balance isThe surface integral is signed: incoming positive electric charge contributes negatively to outward electric current, and outgoing positive electric charge contributes positively. Integrate the charge continuity equation over fixed $V$ and apply the divergence theorem to obtain this balance. Zero net boundary transport gives constant $Q_V$; the global and local statements describe the same conservation law at different scales. ^[ref].

In a steady transport state, charge density and current density are time-independent, so the charge continuity equation reduces to $\nabla \cdot \mathbf{J}=0$. At an electrical circuit junction, choose each terminal electric current $I_k$ positive outward and assume the terminals account for all boundary transport. ThenKirchhoff's current law, ${\sum }_k{I}_k=0$, follows when accumulation of electric charge at the junction is negligible. During storage, retain the same balance with its nonzero derivative; charge conservation continues to hold. This makes charge conservation useful for checking electrical circuit models and for tracking the charging of a capacitor. ^[ref].

The beautiful consistency with Maxwell equations is that the electric field and magnetic field already enforce this transport balance. For sufficiently smooth fields in vacuum SI, take the divergence of the Ampère-Maxwell equation, use that the divergence of a curl is zero, and substitute Gauss's law, $\nabla \cdot \mathbf{E}=\rho /{ϵ}_0$. Constant vacuum permittivity ${ϵ}_0$ givesThe displacement current term supplies the part needed when electric charge accumulates. Historically, Maxwell added that term to extend the steady-current equation; see the existing charge conservation from Maxwell equations for this connection. ^[ref].

A Fourier mode advected with a homologous spherical flow has phase $\mathbf{q}\cdot \mathbf{r}/R(t)$. Its material derivative in the background vanishes because $\dot{\mathbf{k}}=-(\dot{R}/R)\mathbf{k}$. Thus contraction shortens its physical wavelength while preserving its phase on each fluid element. Linearizing the continuity equation and pressureless momentum equation gives $\dot{\delta }=-i\mathbf{k}\cdot {\mathbf{u}}_1$ and ${\dot{\mathbf{u}}}_1+(\dot{R}/R){\mathbf{u}}_1=-i\mathbf{k}{\Phi }_1$. Combining these with the Poisson equation for Newtonian gravity yields $\ddot{\delta }+2(\dot{R}/R)\dot{\delta }-4\pi G\rho \delta =0$ for nonzero compressive wavenumber. It is a local bulk calculation, without the moving cloud-boundary conditions.

For $i{\psi }_t=-D{\nabla }^2\psi +[V(n)+i\Gamma (n)]\psi$, with real $D,V,\Gamma$ and $\psi =\sqrt{n}{e}^{iS}$, the continuity equation isMultiply the equation by ${\psi }^{\ast }$ and subtract its complex conjugate multiplied by $\psi$ to obtain this identity; real interaction and potential terms cancel. Thus the current velocity is $\mathbf{j}/n=2D\nabla S$. With dimensional kinetic operator $-{\hslash }^2{\nabla }^2/(2m)$ and left side $i\hslash {\psi }_t$, this gives the physical superfluid velocity $(\hslash /m)\nabla S$. In normalized equations with $D=1/2$ or $D=1$, the current velocity is respectively $\nabla S$ or $2\nabla S$. The distinction fixes the radial-flow slope of a driven quantum vortex.

Here $\alpha >0$ and $q,p,\gamma$ are real functions of time. Differentiation in position gives ${\psi }_x=[-\alpha (x-q)+ip/\hslash ]\psi$. The standard probability current is thereforeFor a coherent state moving in a quadratic Hamiltonian, the centre follows the classical equation $\dot{q}=p/m$, so the current is the normalized Gaussian probability density transported at its centre velocity. The factor $\sqrt{\alpha /\pi }$ is essential: omitting it changes the current by a dimensional constant and violates the continuity equation for the normalized density. A spatially constant phase does not affect the current.

Let ${\Sigma }_{d0}$ be the uniform background dust surface density of a disk and ${\delta }_d=({\Sigma }_d-{\Sigma }_{d0})/{\Sigma }_{d0}$ its fractional perturbation. Here $k$ is the radial wavenumber, $\Omega$ is the epicyclic frequency and $u$ is a real velocity amplitude. For that uniform background and late velocity $v_x=u\cos (kx-\Omega t)$, linear dust continuity equation gives ${\partial }_t{\delta }_d=-{\partial }_x{v}_x=ku\sin (kx-\Omega t)$. Integrating yields the displayed formula. The coherent travelling-wave fractional excess is $|ku|/\Omega$, with validity requiring this number much smaller than one. The static term $C(x)$ records initial density and transient displacement: drag damps velocity, but supplies no density diffusion. The long-wavelength gas wave has the same fractional density amplitude, so entrainment alone does not produce an additional dust-to-gas ratio enhancement at this order.

For a depth-independent horizontal velocity in a layer of reference depth $H$, the linear transports are $U=Hu$, $V=Hv$. They have dimensions of area per time and satisfy the surface continuity equation ${\eta }_t+U_x+V_y=0$. Their momentum equations are obtained by multiplying the depth-independent velocity equations by $H$.

Diffusion is net transport associated with random microscopic motion. A localized patch of a dilute solute can spread through a stationary solvent: individual molecules move in both directions, but more cross from the crowded side of an imaginary surface than from the less crowded side. This imbalance gives a net diffusive flux down the concentration gradient; the molecules do not need to sense which direction has lower concentration. At uniform concentration, microscopic exchanges continue even though the net diffusive flux vanishes. ^[ref]

The distinction from advection is easiest to see in a dye patch. Uniform bulk motion carries its centre downstream; diffusion broadens it relative to that centre. Both can happen together. A moving or spreading patch by itself does not identify the mechanism: a nonuniform velocity field can also stretch a patch. ^[ref]

For a dilute nonreacting solute in a uniform medium at fixed temperature, Fick's first law models the relative diffusive flux as $\mathbf{j}=-D\nabla C$, with scalar diffusion coefficient $D>0$ of units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$. Here $C$ is the amount of substance per unit volume, so $\mathbf{j}$ has units $\mathrm{mol}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. With carrier velocity field $\mathbf{u}$, the total flux is $C\mathbf{u}+\mathbf{j}$. The continuity equation, with no sources or chemical reactions, givesFor constant $D$ and incompressible flow, this reduces to $C_t+\mathbf{u}\cdot \nabla C=D{\nabla }^{2}C$, where ${\nabla }^2$ is the Laplacian. In a stationary carrier it is the diffusion equation. These assumptions matter: variable $D$ must remain inside the divergence, and changing sources require extra terms. ^[ref]

The useful surprise is the common mathematics: thermal conduction smooths temperature through the analogous heat equation, while molecular diffusion smooths composition. This analogy already motivated the original formulation of Fick's laws. ^[ref] However, the thermal diffusivity and the solutal diffusivity need not be equal. A diffusive time scale $t_D\sim L^2/D$ means that crossing ten times the distance requires roughly one hundred times the time. This is why molecular diffusion is especially effective across thin layers, and why chaotic advection can assist mixing by making those layers thinner. Physical diffusion is a transport mechanism, distinct from the broader probabilistic class of Markov diffusion.

The time derivative of the electric displacement field enters the macroscopic Ampère-Maxwell equation alongside free electric current density. In vacuum, ${\mathbf{J}}_D={ϵ}_0{\partial }_t\mathbf{E}$. It is not transport of free charges through the vacuum gap of a capacitor; its surface integral nevertheless supplies the term needed for compatibility with the continuity equation.

Conservation of probability under a deterministic characteristic flow map with velocity $b(z,t)$ gives the displayed continuity equation on phase space. Along a characteristic, $dP/dt=-P{\nabla }_z\cdot b$. Unlike the Hamiltonian Liouville equation, it allows contraction of phase-space volume and increasing density along trajectories. It still preserves total probability when boundary fluxes vanish.

For $F=(F_1,F_2,F_3)$ in Cartesian coordinates,

For a continuously differentiable vector field, the divergence is the limiting outward flux per unit volume through a small closed surface. Opposite faces of a small rectangular box almost cancel; the remaining differences in the three components give the sum of partial derivatives above. The divergence theorem extends this local balance to a whole region.

For $F=(x,y,0)$, the divergence is $1+1+0=2$. In a planar unit square, the right edge contributes outward flux $1$ and the top contributes $1$, while the left and bottom contribute zero. The total $2$ equals the integral of the planar divergence over area $1$.

If $F$ is a velocity field, positive divergence means local volume expansion, not automatically creation of mass. The continuity equation gives $D\rho /Dt=-\rho \nabla \cdot F$, where $D/Dt$ is the material derivative. For this field, a uniform initial density evolves as $\rho (t)=\rho (0)e^{-2t}$ while material volumes grow as $e^{2t}$. Their product stays constant.

An electric current measures how quickly signed electric charge crosses a chosen fixed oriented surface $S$. Choose its unit normal $\mathbf{n}$; positive electric charge crossing along $\mathbf{n}$ counts positively, while crossing against it counts negatively. If $Q_{\mathrm{cross}}(t)$ is the accumulated signed electric charge transported through $S$, thenHere $\mathbf{J}$ is the current density; the surface integral collects its local normal contributions. Reversing the unit normal reverses $I_S$. The electric current is a signed scalar associated with the whole chosen oriented surface, whereas current density is a vector at each point. The SI unit is the ampere, one coulomb per second. ^[ref]; ^[ref].

For a constant electric current of $2\mathrm{A}$ over a time interval of $3\mathrm{s}$, the transported electric charge is $6\mathrm{C}$. This need not be electric charge accumulating in the wire: in steady transport, electric charge enters and leaves. For a closed oriented surface with outward unit normal, electric charge conservation instead gives $I_{\mathrm{out}}=-d{Q}_{\mathrm{inside}}/dt$. Keeping transported and enclosed electric charge distinct connects electrical circuit bookkeeping to the local continuity equation.

Compare a flat background Friedmann equation with a nearly homogeneous curved region on a slicing of equal local Hubble parameter. To first order in its curvature parameter $K$, subtraction gives $\delta \rho =3K/(8\pi G{a}^2)$, or $\delta =K/(a^2{H}^2)$. Conserved curvature therefore gives $\delta \propto a^2$ in radiation domination and $\delta \propto a$ in matter domination. This construction specifies the slicing: two patches at equal proper time generally do not have the same Hubble parameter. For growing dust perturbations, the continuity equation gives $H_s-H=-\dot{\delta }/3$ to first order. These different time comparisons can share growth powers without sharing the same curvature-to-density coefficient.

For pressureless matter, a positive cosmological constant, flat spatial geometry and negligible radiation, the continuity equation gives ${\rho }_m\propto a^{-3}$. Substitution in the Friedmann equation yields the displayed Hubble parameter, with ${\Omega }_{m0}+{\Omega }_{\Lambda 0}=1$. radiation must be restored at early epochs; the formula does not require matter to be the largest present-day density component.

The four-current combines charge density and current density into the four-vector $J^{\mu }=(c\rho ,\mathbf{J})$. With $x^{\mu }=(ct,\mathbf{x})$, its four-divergence vanishes by the charge continuity equation: ${\partial }_{\mu }{J}^{\mu }={\partial }_t\rho +\nabla \cdot \mathbf{J}=0$. It is the source in the covariant Maxwell equations.

For diffusion with a constant diffusion coefficient $D>0$ and no sources, the model flux is $J=-D\nabla u$. A continuity equation gives $u_t=-\nabla \cdot J=D\Delta u$. This proportionality is a constitutive assumption about transport; the continuity equation expresses conservation. If $D$ varies with position, it must remain inside the divergence, giving $u_t=\nabla \cdot (D\nabla u)$ instead.^[ref]

An exact constructed heat equation example on the whole plane uses dimensionless coordinates with $D=1$ and the smooth initial Gaussian function $u(x,y,0)=e^{-r^2}$. Set $s=1+4t$. For $t\ge 0$,Direct differentiation verifies both the initial condition andThe center decreases, while the region $r^2>s$ increases at that instant. The sign-change circle expands with radius $\sqrt{s}$. A fixed outer location can initially increase and later decrease as this circle passes it.

At time $t$, the peak is $1/s$, and the radius where the profile is $1/e$ times its peak is $\sqrt{s}$. The width grows as the height falls. Integration in polar coordinates givesThus a decreasing peak is compatible with a conserved spatial integral. Dividing $u$ by $\pi$ makes a probability density function with coordinate variances $s/2=1/2+2t$, connecting the same spreading to Brownian motion with diffusion coefficient one. Equivalently the profile is evolved by the Gaussian heat kernel with time parameter $2t$, since that article uses generator $\frac{1}{2}\Delta$.

Use a monotone linear growth factor $b(t)$ as time, $\mathbf{u}=\dot{\mathbf{x}}/\dot{b}$, and $\psi =a\phi /(Cb)$ with $C=3H_0^2{\Omega }_{m0}/2$ and $a_0=1$. The cosmological Poisson equation becomes ${\nabla }^2\psi =\delta /b$; the continuity equation becomes ${\partial }_b\delta +\nabla \cdot [(1+\delta )\mathbf{u}]=0$. The material derivative is $D_b={\partial }_b+\mathbf{u}\cdot \nabla$. Substitution into the pressureless cosmological Euler equation, followed by the growth equation and $\dot{b}=Hbf$, proves the displayed motion equation, with $f=d\ln b/d\ln a$. The density derivative in the conservative continuity equation is Eulerian, not material.