三维空间向量积与旋度算子的高维推广

Vector Products and Curl Operators beyond Three-dimensional Spaces

在欧氏空间中存在数量积和向量积。数量积可定义于任意维数空间,而Massey定理表明满足适当条件的向量积仅存在于三维与七维空间。关于旋度算子,文章得到了平行于Massey定理的结论,即旋度算子也只存在于三维与七维空间,肯定地回答了Math.stackexchange论坛提出的一个问题。并将这一结果与Hurwitz矩阵方程的经典结果以及关于广义Cauchy-Riemann算子的Taussky-Stiefel定理联系起来。

It is generally known that there exists scalar product and vector product in Euclidean space, where scalar product can be defined in any dimension space, while Massey’s theorem shows that vector product only exists in 3-dimensional and 7-dimensional space. In this paper, we obtain the conclusion to the curl operator is parallel to that of Massey’s theorem, that is, the curl operator only exists in 3-dimensional and 7-dimensional space. Furthermore, we give the relationship among our results and the classical result of Hurwitz matrix equation, and Taussky-Stiefel Theorem of generalized Cauchy-Riemann operator.