The problem of inverting the total divergence operator is central to finding components of a given conservation law. This might not be taxing for a low-order conservation law of a scalar partial differential equation, but integrable systems have conservation laws of arbitrarily high order that must be found with the aid of computer algebra. Even low-order conservation laws of complex systems can be hard to find and invert. This paper describes a new, efficient approach to the inversion problem. Two main tools are developed: partial Euler operators and partial scalings. These lead to a line integral formula for the inversion of a total derivative and a procedure for inverting a given total divergence concisely.

求逆全散度算子的问题对于寻找给定守恒定律的组成部分至关重要。对于标量偏微分方程的低阶守恒定律来说，这可能并不费力，但可积系统具有任意高阶的守恒定律，必须借助计算机代数才能找到。即使是复杂系统的低阶守恒定律也很难找到和反转。本文描述了一种新的、有效的反演问题方法。开发了两个主要工具：部分欧拉算子和部分缩放。这些导致了全导数反演的线积分公式和简洁地反演给定总散度的过程。