Solving Differential Equations Involving Vector Fields

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Main Question or Discussion Point

Given the curl and divergence of a vector field, how would one solve for that vector field?

In the particular case I would like to solve, divergence is zero at all coordinates.
 
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jambaugh
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The problem you describe comes up in electromagnetism. The case of zero divergence would be modeled by finding the static magnetic field induced by a steady state electric current density J(x). The curl of the B field is proportional to the current density.

Look up the Biot-Savart's Law which gives the B field as an integral involving the current.
 
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The application here is maxwell's equations. However, my use of point particles precludes the Biot-Savart Law (as the current density is constantly changing)
 

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