An integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensions

While an integration by parts formula for the bilinear form of hypersingular boundary integral operator transient heat equation in three spatial dimensions is available literature, a proof this seems to be missing. Moreover, contains term including time derivative fundamental solution equation, whose interpretation difficult at second glance. To fill these gaps, we provide rigorous general version and alternative representation mentioned term, which valid certain class functions typical tensor-product discretization spaces.