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In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.
 Generally, in particular in the field of electromagnetism and mathematics, flux is usually the integral of a vector quantity, flux density, over a finite surface. It is an integral operator that acts on a vector field similarly to the gradient, divergence and curl operators found in vector analysis. The result of this integration is a scalar quantity called flux.^{[1]} The magnetic flux is thus the integral of the magnetic vector field B over a surface, and the electric flux is defined similarly. Using this definition, the flux of the Poynting vector over a specified surface is the rate at which electromagnetic energy flows through that surface. Confusingly, the Poynting vector is sometimes called the power flux, which is an example of the second usage of flux, below.^{[2]} It has units of watts per square metre (W/m^{2})
In this context, flux has a primary mathematical definition in terms of a surface integral which uses the vectors that represent the force which is causing the flux being studied:
One could argue, based on the work of James Clerk Maxwell,^{[4]} that the transport definition precedes the more recent way the term is used in electromagnetism. The specific quote from Maxwell is "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. It represents the quantity which passes through the surface".
In addition to these few common mathematical definitions, there are many more looser, but equally valid, usages to describe observations from other fields such as biology, the arts, history, and humanities.
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