SIAM Review, Volume 63, Issue 4, Page 854-864, January 2021.
The multivariable chain rule is often challenging to students because it is usually presented with ambiguities and other defects that hamper systematic and reliable application. A very simple formulation combines the derivation operators for functions and for expressions in a manner not found elsewhere due to common confusion between them. Some issues are rooted more deeply than others and are discussed in a broader perspective, starting with the function concept. The approach is illustrated using various applications including the transport equation, partial derivatives of a definite integral, and the distortionless (but not lossless) transmission line. This note is suitable for a lecture in any first-year course covering partial derivatives, as a complement to the other course material.

中文翻译：

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一个精确可靠的多变量链式法则

SIAM 评论，第 63 卷，第 4 期，第 854-864 页，2021 年 1 月。
多变量链式法则通常对学生具有挑战性，因为它通常存在歧义和其他阻碍系统和可靠应用的缺陷。一个非常简单的公式结合了函数和表达式的推导运算符，以一种在其他地方找不到的方式，因为它们之间存在常见的混淆。有些问题比其他问题根深蒂固，并从更广泛的角度进行讨论，从功能概念开始。该方法使用各种应用进行说明，包括传输方程、定积分的偏导数和无失真（但不是无损）传输线。本笔记适用于任何涵盖偏导数的第一年课程的讲座，作为其他课程材料的补充。