Solving Differential Equations Involving Vector Fields

[Savant13]

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.

 

[jambaugh]

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.

 

[Savant13]

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)