# flux of vector field

Let

 $\vec{U}\;=\;U_{x}\vec{i}+U_{y}\vec{j}+U_{z}\vec{k}$

be a vector field in $\mathbb{R}^{3}$  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

 $d\vec{a}\;=\;\vec{n}\,da,$

where $\vec{n}$ is the unit normal vector on the of $da$.

The flux of the vector $\vec{U}$ through the surface $a$ is the

 $\int_{a}\vec{U}\cdot d\vec{a}.$

Remark.  One can imagine that $\vec{U}$ represents the velocity vector of a flowing liquid; suppose that the flow is , i.e. the velocity $\vec{U}$ depends only on the location, not on the time.  Then the scalar product  $\vec{U}\cdot d\vec{a}$ 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  $\vec{U}=x\vec{i}+2y\vec{j}+3z\vec{k}$  and $a$ be the portion of the plane  $x+y+x=1$  in the first octant ($x\geqq 0,\;y\geqq 0,\,z\geqq 0$) with the away from the origin.

One has the constant unit normal vector:

 $\vec{n}\;=\;\frac{1}{\sqrt{3}}\vec{i}+\frac{1}{\sqrt{3}}\vec{j}+\frac{1}{\sqrt% {3}}\vec{k}.$

The flux of $\vec{U}$ through $a$ is

 $\varphi\;=\;\int_{a}\vec{U}\cdot d\vec{a}\;=\;\frac{1}{\sqrt{3}}\int_{a}(x+2y+% 3z)\,da.$

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 $dA$;

 $dA=\cos\alpha\,da$

where $\alpha$ is the angle between the normals of both surface elements, i.e. the angle between $\vec{n}$ and $\vec{k}$:

 $\cos\alpha\;=\;\vec{n}\cdot\vec{k}\;=\;\frac{1}{\sqrt{3}}.$

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

 $\varphi\;=\;\frac{1}{\sqrt{3}}\int_{A}(x+2y+3(1-x-y))\,\sqrt{3}\,dA\;=\;\int_{% 0}^{1}\left(\int_{0}^{1-x}(3-2x-y)\,dy\right)dx\;=\;1$
 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