flux of vector field
be a vector field in and let be a portion of some surface in the vector field. Define one ; if is a closed surface, then the of it. For any surface element of , the corresponding vectoral surface element is
where is the unit normal vector on the of .
The flux of the vector through the surface is the
Remark. One can imagine that represents the velocity vector of a flowing liquid; suppose that the flow is , i.e. the velocity depends only on the location, not on the time. Then the scalar product is the volume of the liquid flown per time-unit through the surface element ; it is positive or negative depending on whether the flow is from the negative to the positive or contrarily.
Example. Let and be the portion of the plane in the first octant () with the away from the origin.
One has the constant unit normal vector:
The flux of through is
However, this surface integral may be converted to one in which is replaced by its projection (http://planetmath.org/ProjectionOfPoint) on the -plane, and is then similarly replaced by its projection ;
where is the angle between the normals of both surface elements, i.e. the angle between and :
Then we also express on with the coordinates and :
|Title||flux of vector field|
|Date of creation||2013-03-22 18:45:25|
|Last modified on||2013-03-22 18:45:25|
|Last modified by||pahio (2872)|
|Synonym||flux of vector|