Flux
A concept describing flow through a surface or substance.
Flux is an idea from applied math and vector calculus, used widely in physics, for any effect that seems to cross or move through a surface or material—regardless of whether anything actually travels. The word comes from Latin *fluxus* ("flow") and *fluere* ("to flow"); Isaac Newton introduced it into calculus as "fluxion."
Joseph Fourier made heat flux central to his analysis of heat transfer. In his *Analytical Theory of Heat*, he defined fluxion as a key quantity and derived expressions for flux based on temperature differences across a slab, then more generally from temperature gradients. James Clerk Maxwell’s work suggests the transport definition of flux came before the electromagnetism definition. Maxwell wrote that to find the quantity passing through a surface, you integrate the flux over every element of that surface—calling the result the surface integral of the flux. Under the transport definition, flux can be a single vector or a vector field (a function of position), and you can integrate it over a surface. Under the electromagnetism definition, flux is already that integral, so integrating it again makes no sense. Maxwell’s quote only fits the transport definition (and assumes flux is a vector field, not a single vector). This is ironic because Maxwell helped develop what we now call electric flux and magnetic flux under the electromagnetism definition. Following his quote, those would be "surface integral of electric flux" and "surface integral of magnetic flux," meaning "electric flux" would be the electric field and "magnetic flux" the magnetic field—implying Maxwell saw these fields as flows. For the electromagnetism definition, "flux density" refers to the derivative along the integrated surface; by the fundamental theorem of calculus, that flux density matches the transport definition. Similarly, electric current (charge per time) has a current density that is a transport flux (charge per time per area). Because the two definitions conflict, and nontechnical English uses flux, flow, and current interchangeably, these terms are often ambiguous.
In transport phenomena (heat transfer, mass transfer, fluid dynamics), flux is the rate of flow of a property per unit area—dimensions [quantity]·[time]⁻¹·[area]⁻¹. The area is the surface the property flows through or across. Examples include water flowing through a river cross-section each second divided by that area, or sunlight energy landing on a patch of ground each second divided by the patch’s area.
Mathematically, the transport definition has three forms, each a special case of the next. The symbol **j** (or J) is used for flux, q for the flowing quantity, t for time, and A for area; vectors are in bold. First, flux as a single scalar: j = I / A, where I = dq/dt (the limit of Δq/Δt as Δt→0). Here the surface is fixed, flat, and the flow is constant and perpendicular to it. Second, flux as a scalar field along a surface—a function of points on the surface: j(**p**) = ∂I/∂A(**p**).
- field
- Applied mathematics, vector calculus, physics
- known_for
- Concept of flux in transport phenomena and electromagnetism; heat flux analysis by Joseph Fourier; contributions by Isaac Newton and James Clerk Maxwell
Lore & Background
The word flux comes from Latin: fluxus means 'flow', and fluere is 'to flow'. As fluxion, this term was introduced into differential calculus by Isaac Newton. The concept of heat flux was a key contribution of Joseph Fourier in the analysis of heat transfer phenomena; his treatise Théorie analytique de la chaleur defines fluxion as a central quantity and derives expressions of flux in terms of temperature differences and gradients. One could argue based on the work of James Clerk Maxwell that the transport definition precedes the definition of flux used in electromagnetism. Maxwell wrote: 'In the case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the surface integral of the flux. Due to conflicting definitions of flux and the interchangeability of flux, flow, and current in nontechnical English, all related terms are sometimes used interchangeably and ambiguously. In transport phenomena, flux is defined as the rate of flow of a property per unit area, with dimensions [quantity]·[time]⁻¹·[area]⁻¹.
Reader's Guide
Flux is a foundational concept bridging mathematics and physics, with two primary definitions that cause ongoing ambiguity. In transport phenomena, flux is a vector quantity describing the magnitude and direction of flow of a substance or property, defined as rate per unit area. In vector calculus, flux is a scalar quantity defined as the surface integral of the perpendicular component of a vector field over a surface. The source highlights that Maxwell's work on electromagnetism used the transport definition conceptually, even as he developed what are now called electric and magnetic flux under the electromagnetism definition. This duality means that terms like flux density and current density can refer to different quantities depending on context. The mathematical framework provides three levels of definition: scalar flux for uniform perpendicular flow through a flat surface, scalar field flux for non-uniform perpendicular flow, and vector field flux for flow in arbitrary directions. These definitions are nested, each a special case of the next. The concept's significance lies in its application across heat transfer, mass transfer, fluid dynamics, and electromagnetism, though the source notes that concrete fluxes in the rest of the article are used according to broad acceptance in the literature regardless of which definition they correspond to.
Did You Know?
- The word flux comes from Latin fluxus meaning 'flow' and fluere meaning 'to flow'.
- Isaac Newton introduced the term 'fluxion' into differential calculus.
- Joseph Fourier's treatise Théorie analytique de la chaleur defined fluxion as a central quantity in heat transfer.
- James Clerk Maxwell's quote on flux implies he conceived of electric and magnetic fields as flows or fluxes of some sort.
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