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If measured radiative flux is $S=A-B>0$ and $S=k{T}^4$ with known $k$, first-order uncertainty propagation gives the displayed result. Independent measurements have zero covariance; strict error bounds combine differently. Even a small fractional background contribution can matter if its relative uncertainty is large. Inferring the covariance or an unknown calibration factor from precision percentages alone is unjustified.

Bedload transport is motion of sediment grains along or close to the bed, represented in a depth-averaged model by a solid-volume flux per unit width. The Exner equation balances this flux divergence and changes in porous-bed solid storage against exchange with suspended sediment. Consistent mass units are necessary when adding this equation to a suspended mass-concentration balance.

Calcium-induced calcium release is activation of release from an intracellular calcium store by cytoplasmic calcium itself. A two-pool model can represent the release flux as a product of saturating functions of store and cytoplasmic concentrations. Together with uptake, leakage and external calcium balance, this positive feedback can generate excitability or sustained oscillation; the existence of oscillation still depends on the full dynamical parameters.

With fixed interior substrate and finite carrier protein, symmetric alternating-access kinetics produce a monotone concave transport curve of the displayed form, with $A,C,D>0$. The flux reverses at equal free concentrations and approaches the finite limit $C/A$ for abundant exterior substrate. Its plateau reflects finite carrier abundance and finite turnover, rather than unlimited transport proportional to the concentration difference. The protein total multiplying the flux is distinct from the total substrate.

A cell-cycle age density counts biological cells per unit time spent in one specified phase of the cell cycle. Its integral over phase age is the population in that phase. Phase age resets on entry into the next phase, so it differs from chronological age since birth. A phase-exit hazard function supplies loss from the phase and incoming flux to the next phase.

Let $F(\zeta ,t)$ map the unit disk conformally to a planar Darcy flow region, with $F(0,t)=0$ at a point source of flux $2\pi Q$. With zero velocity potential on the boundary, the logarithmic source potential becomes $\varphi =Q\log |\zeta |$. Differentiating the inverse map at fixed physical position gives ${\varphi }_t=-Q\mathrm{Re}(F_t/(\zeta F_{\zeta }))$, while $|\nabla \varphi {|}^2=Q^2/|\zeta F_{\zeta }{|}^2$. The boundary law ${\varphi }_t+|\nabla \varphi {|}^2=0$ therefore gives the displayed equation, for nonzero $Q$.

The vector $\zeta F_{\zeta }d\theta$ is outward normal times arclength on $\zeta =e^{i\theta }$. Consequently the equation gives the area growth law $d|D|/dt=2\pi Q$. For a circular map $F=R(t)\zeta$, it reduces to $R\dot{R}=Q$, and hence $R(t{)}^2=R(0{)}^2+2Qt$.

To evaluate outward flux through the lateral surface of a finite right circular cone, add its planar cap and apply the divergence theorem to the enclosed solid. Subtract the cap flux, whose outward normal points away from the solid. For a cone $r=az$ with $0\le z\le h$, the volume element is $rdrd\phi dz$ and the cap has radius $ah$. The cone's vertex has no unique normal, but is a set of area zero; one can remove a small tip and take a limit when the field is regular there.

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.

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.

Let $V$ be a bounded region in Euclidean space with piecewise smooth boundary and outward unit normal vector $n$. For a continuously differentiable vector field $F$ on a neighborhood of its closure,The total outward flux equals the accumulated divergence inside. Contributions across shared internal faces cancel when a region is divided into small pieces.

For $F=(x,y,z)$ on a ball of radius $R$, the divergence is $3$, giving $3(4\pi R^3/3)=4\pi R^3$. On the boundary, $F\cdot n=R$, so the surface integral is also $R(4\pi R^2)=\boxed{4\pi R^3}$. This agreement turns the derivative into a geometric flux measurement. The required regularity matters: a singular field at the center cannot be included without checking its contribution separately.

For positive diffusivity $D(x)$, a diffusion equation with potential-driven drift has diffusive flux $J=-D(c_x+\beta V^{\prime }c)$. On a finite interval with zero flux at both endpoints, the total amount is conserved. The substitution $c=e^{-\lambda t}{e}^{-\beta V}y$ converts the decay eigenvalue problem to the Sturm-Liouville problemThe Rayleigh quotient is $\lambda =\int p|y^{\prime }{|}^2/\int w|y{|}^2\ge 0$. Its zero eigenfunction is constant, giving the stationary solution $c_{\mathrm{eq}}=C{e}^{-\beta V}$. This is a spatial Boltzmann distribution; $D$ changes relaxation rates but cancels out of the zero-flux equilibrium profile. For a sufficiently regular positive $D$ and bounded $V$, the constant eigenfunction is the unique zero mode up to scale.

Physical Review B 80, 115303. doi.org/10.1103/PhysRevB.80.115303

Primary preprint: arxiv.org/abs/0905.1338

Section I discusses edge-channel connectivity and the distinction between fixed-area flux counting and charging-modified periods. Section III.C, Fig. 7, discusses scattering-induced transmission via antidot-bound states. The calculation treats interaction effects and should not be read as a universal prediction of isolated Lorentzian peaks.

Let identical spatially incoherent stellar brightness profiles of equal integrated flux be displaced by projected angles $\pm \beta /2$ along an interferometer baseline. The Fourier shift property makes their normalized signed coherence $g(Y)=g_{\star }(Y)\cos (\pi \beta Y/\lambda )$. The observed visibility is its absolute value. The slow single-star envelope depends on each star's brightness profile, while the oscillatory factor records projected separation. Unequal flux prevents complete cancellation at the binary minima.

For a flux threshold $f_{\min }$ and isotropic point-source luminosity $L$, the detection radius is $r_{\max }=\sqrt{L/(4\pi f_{\min })}$. In a homogeneous Euclidean population the accessible volume is proportional to $L^{3/2}$, so the selected luminosity distribution is proportional to $\phi (L)L^{3/2}$. This assumes flux selection and inverse-square propagation, not an arbitrary surface-brightness-selected diffuse field.

A continuous extension of a vector field across one point need not be differentiable, so it does not automatically meet the classical hypotheses of the divergence theorem. Instead apply the divergence theorem after deleting a ball $B_{\epsilon }$ around that point. If $\Vert \mathbf{F}\Vert \le C{\epsilon }^{\alpha }$ on its boundary in $d$ dimensions, the magnitude of the inner flux is at most a constant times ${\epsilon }^{d-1+\alpha }$. When $d-1+\alpha >0$ this tends to zero. If the divergence is also integrable, passing to the limit recovers the outer flux from the volume integral. For example, a planar vector field with $\Vert \mathbf{F}\Vert \le r$ has an inner flux of order ${\epsilon }^2$, even when its extension is not differentiable at the origin.

In dimensionless Couette-Poiseuille flow in a thin gap with lower-wall velocity one and stationary upper wall, the conserved volume flux satisfies $Q=h/2-h^3{p}_x/12$. Equal pressures at the two ends imply $\int p_{x}dx=0$ and give the displayed expression. It is a weighted average of the positive gap height with weight $h^{-3}$, so $\min h\le 2Q\le \max h$. The local pressure gradient has the sign of $h-2Q$, making the flux constraint useful for locating pressure extrema without solving for the pressure explicitly.

For a nearby Euclidean survey with limiting flux $f_{\min }$, a galaxy of luminosity $L$ is detectable out to $d_{\max }=\sqrt{L/(4\pi f_{\min })}$. Its accessible volume is therefore proportional to $L^{3/2}$. With a Schechter function $\varphi (L)\propto L^{\alpha }{e}^{-L/L_{\ast }}$, counts per logarithmic interval are proportional to $L^{\alpha +5/2}{e}^{-L/L_{\ast }}$ and peak at $(\alpha +5/2)L_{\ast }$ when $\alpha >-5/2$. This selection suppresses faint objects relative to a volume-limited census; finite survey boundaries and cosmological evolution modify the simple law.

On an origin-centered radius-$a$ sphere, $\mathbf{n}=\mathbf{x}/a$. Spherical symmetry gives ${\int }_{S_a}{x}_{i}dS=0$ and ${\int }_{S_a}{x}_i{x}_{j}dS=(4\pi a^4/3){\delta }_{ij}$: off-diagonal moments vanish by reflection, diagonal moments are equal, and their sum is $a^2$ times the surface area. Contracting these identities with $M_{ij}/a$ proves the displayed flux directly, including a zero contribution from a constant field $\mathbf{b}$. It agrees with the divergence theorem, since $\nabla \cdot (M\mathbf{x}+\mathbf{b})=\mathrm{tr}M$.

Let a solid cone of volume $V$ have its planar base in a plane through the origin, and orient its curved surface $C$ outwards. The vector field $\mathbf{F}(\mathbf{x})=\mathbf{x}$ has divergence three. Its flux through the base vanishes because the position vector lies in that plane and is orthogonal to its normal vector. The divergence theorem therefore gives the displayed curved-surface flux. For a circular base of radius $a$ and perpendicular height $b$, this is $\pi a^{2}b$.

For the outward spherical patch $r=R$, $y\le 0$, $z\le 0$, and the polynomial vector field $F=(y{x}^2,x{y}^2-3z^4,x^3+y^2)$, close the quarter-ball with its two planar half-discs. The divergence is $4xy$, whose volume integral vanishes by symmetry in $x$. The caps have fluxes $-3\pi R^6/16$ and $\pi R^4/8$, obtained from half-disc fourth and second moments. The divergence theorem gives the displayed spherical flux. All pieces use normals pointing out of the same closed volume; the caps are auxiliary and must be subtracted.