The Flux of a Vector Field

Vectors are useful for describing the flows of fluids and particles. The central concept that you need is the flux, which is a familiar scalar quantity. The flux of water through a garden hose is the volume of water per unit time that flows through an imaginary plane of unit area that cuts perpendicularly through the hose. For fluid flow, flux has units of (volume of fluid)/[(time)(unit area).  

The dot product of these tw o vectors—the flow velocity vector v and the area element vector ds gives the infinitesimal flux of fluid dj = v-ds through 

EXAMPLE 17.5 Computing the flux. Suppose you have a field of vectors that 

The next section describes Gauss s theorem, a mathematical result that sometimes simplifies the calculation of flux. 

Gauss s Theorem Relates a Volume Property to a Surface Property 

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The divergence operator is the three-dimensional analogue of the differential du of the scalar function u x) of one variable. The analogue of the derivative is the net outflow integral that describes the flux of a vector field across a surface S... 

Gauss s theorem equates the flux of a vector field through a closed surface with the divergence of that same held throughout its volume. This result will be useful in the following chapters. 

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