multi-surface Stokes formula;  multi-region Gaussian formula;  Boundary conditions

For the electromagnetic field field composed of different media, when the divergence source and curl source of the field on the sub-interface are not zero, the field quantity will change on both sides of the sub-interface, and the Stokes formula in the commonly used single medium region is no longer applicable. The Stokes formula in a single medium is generalized, and the multi-curved Stokes formula applicable to the multi-media region is obtained. For the electromagnetic field field composed of different media, a new derivation method for the boundary conditions of electromagnetic field is given by using the generalized multi-surface Stokes formula. The derived formula has wide applications for vector fields composed of multiple media.