If a particular two-dimensional motion is divergenceless, implying that
then the velocity field can be represented in terms of a single
scalar function commonly referred to as the streamfunction. If
we let 
represent the streamfunction, then
and
Eq. (175) is a necessary condition that such a function as implied by Eq. (176) exist. It is readily verified that equations (176) satisfy the condition (175), provided that
which holds as long as possesses first and second
derivatives.