论文标题
N维笛卡尔坐标中的Helmholtz分解和旋转电位
Helmholtz Decomposition and Rotation Potentials in n-dimensional Cartesian Coordinates
论文作者
论文摘要
本文介绍了一种新的方法，可以将Helmholtz分解扩展到N维足够光滑且快速衰减的矢量场。旋转是通过坐标平面内N（N-1）/2旋转的叠加来描述的。源电位和旋转电势是通过将源和旋转密度与拉普拉斯方程的基本溶液进行卷积而获得的。源电势的无旋转梯度和旋转电势总和与原始矢量场的无差旋转。该方法依赖于部分衍生物和牛顿积分，并允许将此标准方法简单地应用于高维矢量字段，而无需使用差分几何和张量计算的概念。
This paper introduces a novel method to extend the Helmholtz Decomposition to n-dimensional sufficiently smooth and fast decaying vector fields. The rotation is described by a superposition of n(n-1)/2 rotations within the coordinate planes. The source potential and the rotation potential are obtained by convolving the source and rotation densities with the fundamental solutions of the Laplace equation. The rotation-free gradient of the source potential and the divergence-free rotation of the rotation potential sum to the original vector field. The approach relies on partial derivatives and Newton integrals and allows for a simple application of this standard method to high-dimensional vector fields, without using concepts from differential geometry and tensor calculus.
