# Divergence free vector field

How can I solve for the vector field $\vec V$, if $\mathrm{div} \vec V=0$? I don't know the general method to find the vector function when we are given its curl or divergence.

• Would Mathematics be a better home for this question? – Qmechanic Aug 25 '17 at 19:43

If it is known that a vector field $\vec V$ is divergence free, that is, $\nabla \cdot \vec V = 0$ which by the fundamental theorem of vector calculus implies the field may be expressed as,
$$\vec V = \nabla \times \vec A$$
where $\vec A$ has the generic term of a vector potential, and in some cases other terminology may be used, such as in fluid dynamics it may be a stream function, for an incompressible fluid flow. This fact follows from the Helmholtz decomposition and the fact that, the divergence of a curl always vanishes. Moreover it is not unique; if $\vec A$ is a valid potential, then,
$$\vec A + \nabla f$$
is also a valid potential, with $f$ a continuously differentiable function. Finally, knowing $\nabla \cdot \vec V=0$ is not sufficient to determine $\vec V$ itself, since there are infinitely many solutions.