Application of a Generalized Leibniz Rule for Calculating Electromagnetic Fields within Continuous Source Regions

In deriving the electric and magnetic fields in a continuous source region by differentiating the vector potential. The central obstacle is the dependence of the integration limits on the differentiation variable. Here, an alternative derivation is presented which evaluates the curl of the vector potential volume integral directly, retaining the dependence of the limits of the differentiation variable. It involves deriving a three-dimensional version of the Leibniz rule for differentiating an integral with variable limits of integration, and using the generalized rule to find the Maxwellian and cavity fields in the source region. A correction term is introduced for each variable integration limit. The extension to a three-dimensional volume integral presented here holds for any function of r and r, and shows that, analogously, one correction term is required per functional limit of the triple integral in each of the derivatives in the curl expression. This generalized three- dimensional Leibniz rule is then tailored to solve the specific problem of determining electromagnetic fields in the source region by direct differentiation of the vector potential.