Domain dependent Dirac's delta and derivatives with an application to electromagnetic boundary integral representations

The domain dependent versions of derivatives and Dirac's delta are defined in distributional sense. These operations enable to obtain domain dependent fundamental solutions and global boundary integral representation formulae. A global representation formula is defined everywhere, also on the boundary, and includes the jump relations of the boundary. The use of the domain dependent objects can be interpreted as taking the boundary limit in prior to integrating by parts when deriving the familiar boundary integral equations. As an application, the representation formulae are obtained for the solutions of the Helmholtz equation and the Maxwell equations.