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Consider [itex]P = \boldsymbol{\nabla} f +\boldsymbol{\nabla}\times \bold{A} [/itex]

where f and A are scalar and vector potentials, respectively, and [itex]P[/itex] is strictly positive and well behaved, and only nonzero in a domain [itex]\mathcal{D}[/itex].

I want to find how the magnitude of

[itex]\int \boldsymbol{\nabla} f dV[/itex] and see how it compares to the size of

[itex]\int P dV[/itex] where we are integrating over all of space [itex]\mathbb{R}^n[/itex] for n the number of dimensions we are working in.

I can use Green's functions to find [itex] f[/itex] in terms of [itex] P[/itex]. I test my results with examples, but when I integrate over all of space to find the magnitude of this term, it appears as if the value I find is dependent on the shape of the integration as I let it go to infinity. This seems non-physical since I have a fixed total input [itex]P[/itex].

Any suggestions would be greatly appreciated.

PS This problem comes from trying to figure out the momentum partitioning between irrotational and rotational fluid flows, if that adds any context.

Cheers,

Nick

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# Helmholtz decomposition and relative magnitudes of irrotational/solenoidal components

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