n维解析向量场的Helmholtz分解和势函数
Helmholtz decomposition and potential functions for n-dimensional analytic vector fields
摘要：
亥姆霍兹分解将一个足够光滑的矢量场分解为梯度场和无发散的旋转场。现有的分解方法对向量场在无穷远处的行为施加了约束，并且需要在整个坐标空间上求解卷积积分。为了允许在$\mathbb{R}^n$中进行亥姆霍兹分解，我们将$\mathbb{R}^3$中的矢量势替换为旋转势，旋转势是描述$n(n-1)/2$在坐标平面内旋转的n维反对称矩阵值映射。给出了Helmholtz分解的三种推导方法：（1）利用$n$维卷积积分求解无穷远衰减场的数值方法;（2）利用线积分求解包括周期函数、指数函数、多元多项式及其线性组合在内的无界增长场的封闭解;（3）所有解析向量场的存在性证明.例子包括Lorenz和Rössler吸引子和具有$n$物种的竞争Lotka-Volterra方程。
英文摘要：
The Helmholtz decomposition splits a sufficiently smooth vector field into a gradient field and a divergence-free rotation field. Existing decomposition methods impose constraints on the behavior of vector fields at infinity and require solving convolution integrals over the entire coordinate space. To allow a Helmholtz decomposition in $\mathbb{R}^n$, we replace the vector potential in $\mathbb{R}^3$ by the rotation potential, an n-dimensional, antisymmetric matrix-valued map describing $n(n-1)/2$ rotations within the coordinate planes. We provide three methods to derive the Helmholtz decomposition: (1) a numerical method for fields decaying at infinity by using an $n$-dimensional convolution integral, (2) closed-form solutions using line-integrals for several unboundedly growing fields including periodic and exponential functions, multivariate polynomials and their linear combinations, (3) an existence proof for all analytic vector fields. Examples include the Lorenz and Rössler attractor and the competitive Lotka-Volterra equations with $n$ species.
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