Flux

According to the transport definition, flux may be a single vector, or it may be a vector field / function of position. In the latter case flux can readily be integrated over a surface. By contrast, according to the electromagnetism definition, flux is the integral over a surface; it makes no sense to integrate a second-definition flux for one would be integrating over a surface twice. Thus, Maxwell's quote only makes sense if "flux" is being used according to the transport definition (and furthermore is a vector field rather than single vector). This is ironic because Maxwell was one of the major developers of what we now call "electric flux" and "magnetic flux" according to the electromagnetism definition. Their names in accordance with the quote (and transport definition) would be "surface integral of electric flux" and "surface integral of magnetic flux", in which case "electric flux" would instead be defined as "electric field" and "magnetic flux" defined as "magnetic field". This implies that Maxwell conceived of these fields as flows/fluxes of some sort.

Given a flux according to the electromagnetism definition, the corresponding flux density, if that term is used, refers to its derivative along the surface that was integrated. By the Fundamental theorem of calculus, the corresponding flux density is a flux according to the transport definition. Given a current such as electric current—charge per time, current density would also be a flux according to the transport definition—charge per time per area. Due to the conflicting definitions of flux, and the interchangeability of flux, flow, and current in nontechnical English, all of the terms used in this paragraph are sometimes used interchangeably and ambiguously. Concrete fluxes in the rest of this article will be used in accordance to their broad acceptance in the literature, regardless of which definition of flux the term corresponds to.

First, flux as a (single) scalar: $$j\; =\; \backslash frac\{I\}\{A\},$$ where $$I\; =\; \backslash lim\_\{\backslash Delta\; t\; \backslash to\; 0\}\backslash frac\{\backslash Delta\; q\}\{\backslash Delta\; t\}\; =\; \backslash frac\{\backslash mathrm\{d\}q\}\{\backslash mathrm\{d\}t\}.$$ In this case the surface in which flux is being measured is fixed and has area A. The surface is assumed to be flat, and the flow is assumed to be everywhere constant with respect to position and perpendicular to the surface.

Second, flux as a scalar field defined along a surface, i.e. a function of points on the surface: $$j(\backslash mathbf\{p\})\; =\; \backslash frac\{\backslash partial\; I\}\{\backslash partial\; A\}(\backslash mathbf\{p\}),$$ $$I(A,\backslash mathbf\{p\})\; =\; \backslash frac\{\backslash mathrm\{d\}q\}\{\backslash mathrm\{d\}t\}(A,\; \backslash mathbf\{p\}).$$ As before, the surface is assumed to be flat, and the flow is assumed to be everywhere perpendicular to it. However the flow need not be constant. q is now a function of p, a point on the surface, and A, an area. Rather than measure the total flow through the surface, q measures the flow through the disk with area A centered at p along the surface.

Finally, flux as a vector field: $$\backslash mathbf\{j\}(\backslash mathbf\{p\})\; =\; \backslash frac\{\backslash partial\; \backslash mathbf\{I\}\}\{\backslash partial\; A\}(\backslash mathbf\{p\}),$$ $$\backslash mathbf\{I\}(A,\backslash mathbf\{p\})\; =\; \backslash underset\{\backslash mathbf\{\backslash hat\{n\}\}\}\{\backslash operatorname\{arg\backslash ,max\}\}\backslash ;\; \backslash mathbf\{\backslash hat\{n\}\}\_\{\backslash mathbf\; p\}\; \backslash frac\{\backslash mathrm\{d\}q\}\{\backslash mathrm\{d\}t\}(A,\backslash mathbf\{p\},\; \backslash mathbf\{\backslash hat\{n\}\}).$$ In this case, there is no fixed surface we are measuring over. q is a function of a point, an area, and a direction (given by a unit vector $\backslash mathbf\{\backslash hat\{n\}\}$), and measures the flow through the disk of area A perpendicular to that unit vector. I is defined picking the unit vector that maximizes the flow around the point, because the true flow is maximized across the disk that is perpendicular to it. The unit vector thus uniquely maximizes the function when it points in the "true direction" of the flow. (Strictly speaking, this is an abuse of notation because the "argmax" cannot directly compare vectors; we take the vector with the biggest norm instead.)

If the flux j passes through the area at an angle θ to the area normal $\backslash mathbf\{\backslash hat\{n\}\}$, then the dot product $$\backslash mathbf\{j\}\; \backslash cdot\; \backslash mathbf\{\backslash hat\{n\}\}\; =\; j\backslash cos\backslash theta.$$ That is, the component of flux passing through the surface (i.e. normal to it) is jcos θ, while the component of flux passing tangential to the area is jsin θ, but there is no flux actually passing through the area in the tangential direction. The only component of flux passing normal to the area is the cosine component.

For vector flux, the surface integral of j over a surface S, gives the proper flowing per unit of time through the surface: $$\backslash frac\{\backslash mathrm\{d\}q\}\{\backslash mathrm\{d\}t\}\; =\; \backslash iint\_S\; \backslash mathbf\{j\}\; \backslash cdot\; \backslash mathbf\{\backslash hat\{n\}\}\backslash ,\; dA\; =\; \backslash iint\_S\; \backslash mathbf\{j\}\; \backslash cdot\; d\backslash mathbf\{A\},$$ where A (and its infinitesimal) is the vector area combination $\backslash mathbf\{A\}\; =\; A\; \backslash mathbf\{\backslash hat\{n\}\}$ of the magnitude of the area A through which the property passes and a unit vector $\backslash mathbf\{\backslash hat\{n\}\}$ normal to the area. Unlike in the second set of equations, the surface here need not be flat.

Finally, we can integrate again over the time duration t₁ to t₂, getting the total amount of the property flowing through the surface in that time ( t₂ −  t₁): $$q\; =\; \backslash int\_\{t\_1\}^\{t\_2\}\backslash iint\_S\; \backslash mathbf\{j\}\backslash cdot\; d\backslash mathbf\; A\backslash ,\; dt.$$

1. Momentum flux, the rate of transfer of momentum across a unit area (N·s·m⁻²·s⁻¹). (Newton's law of viscosity)
   (1978). 9780719533822, John Murray. ISBN 9780719533822
2. Heat flux, the rate of heat flow across a unit area (J·m⁻²·s⁻¹). (Fourier's law of conduction)
   H.S. Carslaw (2025). 9780198533030, Oxford University Press. ISBN 9780198533030
   (This definition of heat flux fits Maxwell's original definition.)
3. Diffusion flux, the rate of movement of molecules across a unit area (mol·m⁻²·s⁻¹). (Fick's law of diffusion)
4. Volumetric flux, the rate of volume flow across a unit area (m³·m⁻²·s⁻¹). (Darcy's law of groundwater flow)
5. Mass flux, the rate of mass flow across a unit area (kg·m⁻²·s⁻¹). (Either an alternate form of Fick's law that includes the molecular mass, or an alternate form of Darcy's law that includes the density.)
6. Radiative flux, the amount of energy transferred in the form of photons at a certain distance from the source per unit area per second (J·m⁻²·s⁻¹). Used in astronomy to determine the magnitude and spectral class of a star. Also acts as a generalization of heat flux, which is equal to the radiative flux when restricted to the electromagnetic spectrum.
7. Energy flux, the rate of transfer of energy through a unit area (J·m⁻²·s⁻¹). The radiative flux and heat flux are specific cases of energy flux.
8. Particle flux, the rate of transfer of particles through a unit area (number m⁻²·s⁻¹)

These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the divergence of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. For incompressible flow, the divergence of the volume flux is zero.

This flux has units of mol·m⁻²·s⁻¹, and fits Maxwell's original definition of flux.