Abstract: This paper expounds though the line integral that the vector field along the first-order infinitesimal line is also the first-order dimensionless，but in particular cases demand the computational accuracy must achieve the second-order dimensionless. For this purpose，we need to divide the first-order infinitesimal line segment into infinitely many second-order infinitesimal line element to calculate the line integral，and give the law that followed by the result that obtained through such calculation: the theorem of vector field along first-order infinitesimal line segment’s line integral under second-order infinitesimal accuracy. We introduce application of this theorem in curl theory and in the theory of electromagnetic field tangential boundary value relations.

Key words: the first-order infinitesimal line segment, line integral, the second-order infinitesimal accuracy, curl theory, theory of electromagnetic field tangential boundary value relations