Abstract: It is introduced the concept of absolute integral in Riemannian spaces and in fibre bundle space (Gauge field),with respect to frames at every point.This is just the inverse of the absolute differential. After discussing the exterior differential d(dxi)≠0,rewrote the exterior differential form into symmetric form and established the relation between the exterior differential form and the absolute differential.By the aid of the absolute integral,it is improved Stokes’ formula:The strict definitions of the circulation,the curl and the divergence were obtained (usually,they were obtained only by analogy in Euclid-space,unable to apply here).It had been proved that the curvature tensor is a curl(grad),not zero except in Eucldean space,and so discovered the essence of Bianchi identity:div(rot(grad))=0,the curvature,forming tube field,is invariant along the tube,i.e.pointed out that Gauge fields are curl fields of gradients and so on.By the way,it is obtained the torsion tensor is rot of base of frame also.