Transport Theorem for Spaces and Subspaces of Arbitrary Dimensions

Using the apparatus of traditional differential geometry, the transport theorem is derived for the general case of a M-dimensional domain moving in a N-dimensional space, M ≤ N . The interesting concepts of curvatures and normals are illustrated with well-known examples of lines, surfaces and volumes. The special cases where either the space or the moving subdomain are material are discussed. Then, the transport at hypersurfaces of discontinuity is considered. Finally, the general local balance equations for continuum of arbitrary dimensions with discontinuities are derived.