Let
be a vector field in
where
The flux of the vector
through the surface
is the surface integral
Remark. One can imagine that
represents the velocity
vector of a flowing liquid; suppose that the flow is stationary, 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 side to the positive side or contrarily.
Example. Let
and
be the portion of the plane
in the first octant (
) with the positive normal away from the origin.
One has the constant unit normal vector:
The flux of
However, this surface integral may be converted to one in which
is replaced by its projection
on the
-plane, and
is then similarly replaced by its projection
;
where
Then we also express
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