A method of spatial (integral) differentiation of multivector fields in an N
-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.