New product differentiation rule for paired scalar reciprocal functions
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When integral kernel of an integral transform is being formed, it should be the outcome of scalar product differentiation rule if the kernel is supposed to be eventually used as an integrand in a prospective integration. Yet it has already been shown that despite ensuing from properly performed differentiation, the resulting integral kernel contains, beside the covariant differential that is suitable for integration, also a certain contravariant term, which is not appropriate for integration in the same space as the covariant differential. But the contravariant term also can be turned into proper, though multiplicatively inverse covariant differential, if placed within a space that is reciprocal to the given primary space in which the first, covariant differential, is represented naturally. This uncharacteristic conversion of the contravariant expression from the primary space into the reciprocal covariant differential in the dual reciprocal space that is paired with the given primary space, can be considered as indirect proof that pairing of mutually dual reciprocal spaces is necessary in order to properly form operationally legitimate and geometrically valid differential structures. Consequently, the pairing of an infinitesimal descending singularity of the 2D domain of complex numbers with an infinitely ascending singularity deployed in the 1D domain of real numbers requires certain dual reciprocal spatial or quasispatial structures, for the downward transition from 2D descending complex singularities to the 1D ascending “real” singularities to be meaningfully/unambiguously implemented. Furthermore, just as integration by parts formula is a counterpart of the regular product differentiation rule, a new multispatial scalar product integration rule is proposed as a counterpart to the singlespatial product differentiation rule, and introduced by analogy to the latter, “regular” product integration rule.
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