矢量基尔霍夫公式的证明过程中存在的纰漏及严格证明

Error in Vector Kirchhoff Formula Proof Process and Strict Proof Method

揭示了在各类经典著作中,矢量基尔霍夫公式证明过程中具示范性、普遍性的错误,对其来源、特征进行了剖析,理清了证明的线索,突破了证明过程中的核心障碍.指出:1)矢量基尔霍夫公式的成立条件是被积函数在积分区域上具有连续二阶偏导数;2)作为一个积分定理,其证明无法直接在微分尺度进行.同时,矢量微分算符在自然坐标系中的表达须特别注意基矢选择及变换,尤其针对积分曲面与等势面的法向量.

Vector Kirchhoff formula is the basis of the optical diffraction theory, but the certification and understanding of it is not flawless in all kinds of classic works. Although such flaws are analyzed by some researchers in the literature available, but such analyses are not complete. Mainly reflected in: 1. The Stratton-Chu formula in integral surface normals base vector processing problems: 1) it’s misused as a constant vector, 2) although it is treated as a variable, it fails to consider the geometrical constraints of the surface itself, resulting in the calculation error. 2. The error of the differential operator in the natural coordinate system is to confuse the "equipotential surface" of the integral surface and the integrand, which will cause the expression of the method to the base vector;3. The transition of the original form of vector Sommerfeld radiation condition to the scalar form is not proved, and abandoning the two related to another argument lacks the physical connotation. These issues shows us in on the source model involving the surface integral vector analysis must pay attention to several points: in the same surface integral problem must strictly distinguish between the integrand equipotential surface normal vector and normal vector integral surface;The expression of the vector differential operator in the natural coordinate system must pay special attention to the selection and transformation of the base vector. We draw two conclusions through proof: 1. the vector kirchhoff formula is established if the integrand has a continuous second partial derivative in the integral region;2.(vector kirchhoff formula) as an integral theorem, just on the analysis of the differential scale unable to provide complete and correct. At the same time, firmly grasp the physical image, this paper starting from the special case analysis, combining tensor operations, proved by mistake put item details, in the process of processing by integral and gives the strict proof of vector kirchhoff formula, therefore, this paper has some demonstration.