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A version of the Helmholtz theorem says that, under opportune assumptions on the vector field $\boldsymbol{F}:\mathbb{R}^3\to\mathbb{R}^3$ and on $V\subset\mathbb{R}^3$ the following identity holds: $$\boldsymbol{F}(\boldsymbol{x})=\frac{1}{4\pi} \nabla\times\int_V \frac{\nabla'\times\boldsymbol{F}(\boldsymbol{x}')}{\|\boldsymbol{x}-\boldsymbol{x}'\|}d\mu' -\frac{1}{4\pi} \nabla\times\int_{\partial V} \frac{\hat{\boldsymbol{n}}(\boldsymbol{x'})\times\boldsymbol{F}(\boldsymbol{x}')}{\|\boldsymbol{x}-\boldsymbol{x}'\|}d\sigma'$$$$-\frac{1}{4\pi}\nabla \int_V \frac{\nabla'\cdot\boldsymbol{F}(\boldsymbol{x}')}{\|\boldsymbol{x}-\boldsymbol{x}'\|}d\mu' +\frac{1}{4\pi}\nabla\int_{\partial V} \frac{\boldsymbol{F}(\boldsymbol{x}')\cdot\hat{\boldsymbol{n}}(\boldsymbol{x'})}{\|\boldsymbol{x}-\boldsymbol{x}'\|} d\sigma' .$$

I think (hope) I have proved (here) to myself the validity of such a decomposition for $\boldsymbol{F}\in C^2(\mathbb{R}^3)$ compactly supported.

Are such assumptions consistent with the physical applications of the theorem?

I suspect that $\boldsymbol{F}$ usually is reasonably supposed to be null outside a bounded domain in application of the decomposition to physics, but I would be very grateful to anybody giving a knowledgeable answer on the issue.

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In continuum mechanics, you can apply the theorem derived for compact support field to velocity of finite-sized body like raindrop. All realistic models involve finite-sized fluid bodies, so the assumption about compact support is valid there.

In electromagnetic theory, fields generally extend to infinity and do not have compact support. Static fields decay as $1/r^2$, but generally fields decay as slow as $1/r$ if accelerated charged particles were present for the infinite past time. So to apply the decomposition to such fields confidently, one should be able to give an explanation that does not depend on the field having compact support, or limit oneself to fields that are finite only in a limited region of space and then drop to zero (which is not very realistic but may work in some practical calculations).

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