Editing Stream function (section)

=== Relation to streamlines ===

Consider two-dimensional plane flow with two infinitesimally close points <math>P = (x,y,z)</math> and <math>P ' = (x+dx,y+dy,z)</math> lying in the same horizontal plane. From calculus, the corresponding infinitesimal difference between the values of the stream function at the two points is

:<math>\begin{align}
\mathrm{d} \psi (x, y, t) &= \psi (x + \mathrm{d} x, y + \mathrm{d} y, t) - \psi(x, y, t) \\
&= {\partial \psi \over \partial x} \mathrm{d} x + {\partial \psi \over \partial y} \mathrm{d} y \\
&= \nabla \psi \cdot \mathrm{d} \mathbf{r}
\end{align}
</math>

Suppose <math>\psi</math> takes the same value, say <math>C</math>, at the two points <math>P</math> and <math>P '</math>. Then this gives

:<math>
0 = \nabla \psi \cdot \mathrm{d} \mathbf{r} ,
</math>

implying that the vector <math>\nabla \psi</math> is normal to the surface <math>\psi = C</math>. Because <math>\mathbf{u} \cdot \nabla \psi = 0</math> everywhere (e.g., see [[Stream function#Condition of existence|In terms of vector rotation]]), each streamline corresponds to the intersection of a particular stream surface and a particular horizontal plane. Consequently, in three dimensions, unambiguous identification of any particular streamline requires that one specify corresponding values of both the stream function and the elevation (<math>z</math> coordinate).

The development here assumes the space domain is three-dimensional. The concept of stream function can also be developed in the context of a two-dimensional space domain. In that case level sets of the stream function are curves rather than surfaces, and streamlines are level curves of the stream function. Consequently, in two dimensions, unambiguous identification of any particular streamline requires that one specify the corresponding value of the stream function only.