In fluid dynamics, the stream function is defined for two-dimensional flows. It is the flux across a line (any line!) between a given reference point and the point at which the value of the function is desired.

Note that since streamlines are tangent to the flow, the value of the stream function must be the same along a streamline. If there were a flux across a line, it would necessarily not be tangent to the flow, hence would not be a streamline.

The usefulness of the stream function lies in the fact that the velocity components in the x- and y- directions at a given point are given by the partial derivatives of the stream function at that point.