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Deriving the PDE for a vector field from its curl and divergence

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How do they come up with the equations in (308) mathematically? Why do (308) give solutions to (285) and (286). Or why do (308) determine whether (285) and (286) have 1 or more solutions? I don't wonder about the proof for why the La place (309) introduced as a general equation later in the text has only 1 solution. Thanks

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Can we determine a vector field given just its curl, divergence and physical boundary conditions and is this determination unique?
Let F and A be vector fields, and is a scalar function such that:
If we want to write these equations explicitly in Cartesian coordinates we get for equations. The first is the scalar equation:
And three from the vector equation (1.2) when we equate components on both sides:

The whole point of the exercise is to find a vector field F that will satisfy these four first-order partial differential equations, ...

Solution Summary

The solution shows how to decouple the first order differential equations arising from the curl and divergence and turn them into Helmholtz equations and show what are the conditions for uniqueness.