Proof for the Directivity of Circulation Intensity of Vector Field

Abstract

At present, the introduction of curl is based on the directivity of circulation intensity, which means that circulation intensity under any given normal direction can be expressed as the scalar product between a vector (that is, the curl) and the unit vector in the normal direction. However, there is usually no enough explanation on the validity of the directivity characteristics. Beginning from the definition of circulation intensity, a detailed deduction procedure of the expression of circulation intensity was present under any normal direction by using a square integral path, and the validation of the directivity characteristics was verified. The work in this paper provides students with a feasible mathematical approach to understand the directivity characteristics, and can also be treated as a comprehensive exercise for vector analysis and field theory.