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• W3100182589 abstract "Abstract Any square-integrable vector field $$mathbf {f}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>f</mml:mi> </mml:math> over a sphere $$mathbb {S}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>S</mml:mi> </mml:math> can be decomposed into three unique contributions: one being the gradient of a function harmonic inside the sphere (denoted by $$mathbf {f}_+$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:math> ), one being the gradient of a function harmonic in the exterior of the sphere (denoted by $$mathbf {f}_-$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>f</mml:mi> <mml:mo>-</mml:mo> </mml:msub> </mml:math> ), and one being tangential and divergence-free (denoted by $$mathbf {f}_{df}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mi>df</mml:mi> </mml:mrow> </mml:msub> </mml:math> ). In geomagnetic applications this is of relevance because, if we consider $$mathbf {f}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>f</mml:mi> </mml:math> to be identified with a magnetization, only the contribution $$mathbf {f}_+$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:math> can generate a non-vanishing magnetic field in the exterior of the sphere. Thus, we call $$mathbf {f}_-$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>f</mml:mi> <mml:mo>-</mml:mo> </mml:msub> </mml:math> and $$mathbf {f}_{df}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow> <mml:mi>df</mml:mi> </mml:mrow> </mml:msub> </mml:math> “silent” and $$mathbf {f}_+$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:math> “nonsilent”. If $$mathbf {f}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>f</mml:mi> </mml:math> is known to be spatially localized in a subregion of the sphere, then $$mathbf {f}_+$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:math> and $$mathbf {f}_-$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>f</mml:mi> <mml:mo>-</mml:mo> </mml:msub> </mml:math> are coupled due to their potential field nature. In this short paper, we derive an approach that makes use of this coupling in order to compute the contribution $$mathbf {f}_-$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>f</mml:mi> <mml:mo>-</mml:mo> </mml:msub> </mml:math> from knowledge of the contribution $$mathbf {f}_+$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>f</mml:mi> <mml:mo>+</mml:mo> </mml:msub> </mml:math> ." @default.
• W3100182589 created "2020-11-23" @default.
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• W3100182589 date "2020-09-18" @default.
• W3100182589 modified "2023-10-13" @default.
• W3100182589 title "A brief note on the computation of silent from nonsilent contributions of spatially localized magnetizations on a sphere" @default.
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• W3100182589 doi "https://doi.org/10.1007/s13137-020-00161-z" @default.
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