View source for PlanetPhysics/Flux of Vector Field

Let
<math>\vec{U} \;=\; U_x\vec{i}+U_y\vec{j}+U_z\vec{k}</math>
be a [[../NeutrinoRestMass/|vector field]] in <math>\mathbb{R}^3</math>\, and let <math>a</math> be a portion of some surface in the vector field.\, Define one side of <math>a</math> to be positive; if <math>a</math> is a closed surface, then the positive side must be the outer surface of it.\, For any surface element <math>da</math> of <math>a</math>, the corresponding ''vectoral surface element''  is
<math>d\vec{a} \;=\; \vec{n}\,da,</math>
where <math>\vec{n}</math> is the unit normal [[../Vectors/|vector]] on the positive side of <math>da</math>.

The ''[[../AbsoluteMagnitude/|flux]]'' of the vector <math>\vec{U}</math> through the surface <math>a</math> is the surface integral
<math>\int_a\vec{U} \cdot d\vec{a}.</math>\\

'''Remark.''' \, One can imagine that <math>\vec{U}</math> represents the [[../Velocity/|velocity]] vector of a flowing liquid; suppose that the flow is stationary, i.e. the velocity <math>\vec{U}</math> depends only on the location, not on the time.\, Then the [[../DotProduct/|scalar product]] <math>\vec{U} \cdot d\vec{a}</math> is the [[../Volume/|volume]] of the liquid flown per time-unit through the surface element <math>da</math>; it is positive or negative depending on whether the flow is from the negative side to the positive side or contrarily.

'''Example.''' \, Let\, <math>\vec{U} = x\vec{i}+2y\vec{j}+3z\vec{k}</math>\, and <math>a</math> be the portion of the plane \,<math>x+y+x = 1</math>\, in the first octant (<math>x \geqq 0,\; y \geqq 0,\, z \geqq 0</math>) with the positive normal away from the origin.

One has the constant unit normal vector:
<math>\vec{n} \;=\; \frac{1}{\sqrt{3}}\vec{i}+\frac{1}{\sqrt{3}}\vec{j}+\frac{1}{\sqrt{3}}\vec{k}.</math>
The flux of <math>\vec{U}</math> through <math>a</math> is
<math>\varphi \;=\; \int_a\vec{U}\cdot d\vec{a} \;=\; \frac{1}{\sqrt{3}}\int_a(x+2y+3z)\,da.</math>

However, this surface integral may be converted to one in which <math>a</math> is replaced by its projection <math>A</math> on the <math>xy</math>-plane, and <math>da</math> is then similarly replaced by its projection <math>dA</math>;
<math>dA = \cos\alpha\, da</math>
where <math>\alpha</math> is the angle between the normals of both surface elements, i.e. the angle between <math>\vec{n}</math> and <math>\vec{k}</math>:
<math>\cos\alpha \;=\; \vec{n}\cdot\vec{k} \;=\; \frac{1}{\sqrt{3}}.</math>
Then we also express <math>z</math> on <math>a</math> with the coordinates <math>x</math> and <math>y</math>:
<math>\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</math>

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