A method of spatial (integral) differentiation of multivector fields in anN -dimensional manifold M, into which a hyper-rectangle [a, b] is mapped by a bijective smooth map r : [a, b] → M, has been introduced. For a class of discontinuous multivector fields a new concept of a residual field as well as the concept of total Hn integrability have been defined. Finally, this led naturally to an extension of Cauchy's residue theorem in M.