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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.

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  • $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$
    – Qmechanic
    Aug 25, 2017 at 19:43

1 Answer 1

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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.

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  • $\begingroup$ I'm supposed to find the radial and azimuthal velocities in cylindrical coordinates. So I wrote down the expression of div V in cylindrical coordinates. I don't know how to proceed after that. $\endgroup$
    – Aparimitha
    Aug 27, 2017 at 13:51

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