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, Yaghjian (1985) explains that the central obstacle is the dependence of the integration limits on the differentiation variable. Since it is not mathematically rigorous to assume the curl and integral signs are interchangeable, he uses an integration variable substitution to circumvent this problematic dependence. Here, we present an alternative derivation, which evaluates the curl of the vector potential volume integral directly, retaining the dependence of the limits of integration on the differentiation variable. It involves deriving a three-dimensional version of 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.