flux of vector field



be a vector field in 3  and let a be a portion of some surface in the vector field.  Define one ; if a is a closed surface, then the of it.  For any surface element da of a, the corresponding vectoral surface element is


where n is the unit normal vector on the of da.

The flux of the vector U through the surface a is the


Remark.  One can imagine that U represents the velocity vector of a flowing liquid; suppose that the flow is , i.e. the velocity U depends only on the location, not on the time.  Then the scalar productMathworldPlanetmath Uda is the volume of the liquid flown per time-unit through the surface element da; it is positive or negative depending on whether the flow is from the negative to the positive or contrarily.

Example.  Let  U=xi+2yj+3zk  and a be the portion of the plane  x+y+x=1  in the first octant (x0,y0,z0) with the away from the origin.

One has the constant unit normal vector:


The flux of U through a is


However, this surface integral may be converted to one in which a is replaced by its projection (http://planetmath.org/ProjectionOfPoint) A on the xy-plane, and da is then similarly replaced by its projection dA;


where α is the angle between the normals of both surface elements, i.e. the angle between n and k:


Then we also express z on a with the coordinatesMathworldPlanetmathPlanetmath x and y:

φ=13A(x+2y+3(1-x-y))3𝑑A=01(01-x(3-2x-y)𝑑y)𝑑x= 1
Title flux of vector field
Canonical name FluxOfVectorField
Date of creation 2013-03-22 18:45:25
Last modified on 2013-03-22 18:45:25
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Definition
Classification msc 26B15
Classification msc 26B12
Synonym flux of vector
Related topic GaussGreenTheorem
Related topic MutualPositionsOfVectors
Related topic AngleBetweenTwoVectors
Defines flux