The operation of the exponential of the *curl* on a general vector field is studied. It is nonlocal, and it is shown that the resultant field may be written as the sum of integrals over all space. The formal solutions of the time-dependent Maxwell’s equations for an arbitrary current density are first written in terms of the *curl*, and explicit expressions for the electric and magnetic fields are given in terms of the source current densities loaded with these kernels. This method of deriving the fields obviates the introduction of electromagnetic potentials. The well-known expressions for the fields are derived for a fixed oscillating dipole and a charge in arbitrary motion. For a moving dipole the source current includes, in addition to the polarization current, a coupling of the polarization to the velocity of the dipole known as the Röntgen current. The magnetic field due to this current is given.

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