Editing Stream function (section)

==== Flux through the test surface ====
[[File:Stream function definition.svg|thumb|right|The volume [[flux]] through the test surface connecting the points <math>A</math> and <math>P.</math>]]
The total [[flux|volumetric flux]] through the test surface is
:<math>
Q (x, y, t) = \int_0^b \int_0^L \mathbf{u} \cdot \hat\mathbf{n}\, \mathrm{d}s\, \mathrm{d}z
</math>
where <math>s</math> is an arc-length parameter defined on the curve <math>AP</math>, with <math>s = 0</math> at the point <math>A</math> and <math>s = L</math> at the point <math>P</math>.
Here <math>\hat\mathbf{n}</math> is the unit vector perpendicular to the test surface, i.e.,
:<math>
\hat\mathbf{n}\, \mathrm{d}s = -R\, \mathrm{d} \mathbf{r} = \begin{bmatrix}
\mathrm{d}y \\
- \mathrm{d}x \\
0
\end{bmatrix}
</math>
where <math>R</math> is the <math>3 \times 3</math> [[rotation matrix]] corresponding to a <math>90^\circ</math> anticlockwise rotation about the positive <math>z</math> axis:
:<math>
R = R_z(90^\circ) = \begin{bmatrix}
0 & -1 & 0 \\  
1 & 0 & 0 \\
0 & 0 & 1 
\end{bmatrix}.
</math>
The integrand in the expression for <math>Q</math> is independent of <math>z</math>, so the outer integral can be evaluated to yield
:<math>
Q (x, y, t) = b\, \int_A^P \left( u\, \mathrm{d} y - v\, \mathrm{d} x \right)
</math>