Editing Stream function (section)

=== In terms of vector potential and stream surfaces ===
In general, a divergence-free field like <math>\mathbf{u}</math>, also known as a [[solenoidal vector field]], can always be represented as the curl of some [[vector potential]] <math>\boldsymbol{A}</math>:
:<math>
\mathbf{u}= \nabla \times  \boldsymbol{A}.
</math>

The stream function <math>\psi</math> can be understood as providing the strength of a vector potential that is directed perpendicular to the plane:<ref>{{cite book|doi=10.1016/B978-0-12-815489-2.00005-8 |chapter=Viscous Fluid Flow |title=Free-Surface Flow |date=2019 |last1=Katopodes |first1=Nikolaos D. |pages=324–426 |isbn=978-0-12-815489-2 }}</ref>
:<math>
\boldsymbol{A}(x,y,t) = \begin{bmatrix}
0 \\
0 \\
\psi(x,y,t)
\end{bmatrix},
</math>
in other words <math>\boldsymbol{A} = \psi \hat\mathbf{z}</math>, where <math>\hat\mathbf{z}</math> is the unit vector pointing in the positive <math>z</math> direction.

This can also be written as the vector cross product
:<math>
\mathbf{u} = \nabla \psi \times \hat\mathbf{z}
</math>
where we've used the [[vector calculus identity]]
:<math>
\nabla \times \left( \psi \hat\mathbf{z} \right) = \psi \nabla \times \hat\mathbf{z} + \nabla \psi \times \hat\mathbf{z}.
</math>
Noting that <math>\hat\mathbf{z} = \nabla z</math>, and defining <math>\phi = z</math>, one can express the velocity field as
:<math>
\mathbf{u} = \nabla \psi \times \nabla \phi .
</math>
This form shows that the level surfaces of <math>\psi</math> and the level surfaces of <math>z</math> (i.e., horizontal planes) form a system of orthogonal [[Streamsurface|stream surfaces]].