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Vector-fields are vector valued functions which define a vector at each point in space. Examples of the vector field include the electric field and the velocity of a fluid.
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Solenoidal fields in $D$ spatial dimensions
In the T. Sakurai paper cited below, page 209, the author discusses how the magnetic field can be written in terms to the cross-product
$$ \vec B = \vec\nabla \chi \times \vec\nabla \eta,$$
for two …