Flux

:This article is about the concept of flux in science and mathematics. For other uses of the word, see flux (disambiguation).

Meaning of flux and theorems

An example of a flux is the magnitude of a river's current, that is, the amount of water that flows through a cross-section of the river each second. The amount of sunlight that lands on a patch of ground each second is also a kind of flux.

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To better understand the concept of flux, imagine a butterfly net. The amount of air moving through the net at any given instant in time is the flux. If the wind speed is high, then the flux through the net is large. If the net is made bigger, then the flux would be larger even though the wind speed is the same. For the most air to move through the net, the opening of the net must be facing the direction the wind is blowing. If the net opening is parallel to the wind, then no wind will be moving through the net.

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As a mathematical concept, flux is represented by the surface integral of a vector field,

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:Phi_f = int_S mathbf{F} cdot mathbf{dA}

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where F is a vector field, dA is the area element of the surface S, directed as the surface normal, and Phi_f  is the resulting flux.

Related Topics:
Vector field - Area - Surface normal

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The surface has to orientable, i.e. two sides can be distinguished: the surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative.

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The surface normal is directed accordingly, usually by the right-hand rule.

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Conversely, one can consider the flux the more fundamental quantity, and call the vector field the flux density.

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Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive divergence (sources) and end at areas of negative divergence (sinks).

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See also the image at right: the number of red arrows passing through a unit area is the flux density, the curve encircling the red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the inner product of the vector field with the surface normals.

Related Topics:
Curve - Inner product

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If the surface enclosed a 3D region, usually the surface is oriented such that the outflux is counted positive; the opposite is the influx.

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The divergence theorem states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the divergence).

Related Topics:
Divergence theorem - Divergence

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If the surface is not closed, it has an oriented curve as boundary. Stokes theorem states that the flux of the curl of a vector field is the path integral of the vector field over this boundary. This path integral is also called circulation, especially in fluid dynamics. Thus the curl is the circulation density.

Related Topics:
Stokes theorem - Curl - Path integral - Circulation

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We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas.

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