The paper presents analyses of multivariable calculus learning-teaching phenomena through the lenses of DNR-based instruction, focusing on several foundational calculus concepts, including cross product, linearization, total derivative, Chain Rule, and implicit differentiation. The analyses delineate the nature of current multivariable calculus instruction and theorize the potential consequences of the two types of instruction to student learning. The following are among the consequences revealed: understanding the concept of function as a covariational processes, and the concept of derivative as a linear approximation; understanding the relation between composition of functions and their Jacobian matrices; understanding the idea underlying the concept of parametrization and the rationale underpinning the process of implicit differentiation; thinking in terms of structure, appending external inputs into a coherent mental representation, and making logical inferences.

本文通过基于DNR的教学视角，对多变量微积分学习教学现象进行了分析，重点研究了几种基本的微积分概念，包括叉积，线性化，总导数，链式规则和隐式微分。这些分析描述了当前多变量微积分教学的性质，并对两种类型的教学对学生学习的潜在后果进行了理论推论。结果揭示了以下几点：将函数的概念理解为协变过程，将导数的概念理解为线性近似；了解函数的组成及其雅可比矩阵之间的关系; 了解潜在的概念隐性分化过程的参数化和基本原理；在结构方面进行思考，将外部输入附加到连贯的心理表示中，并进行逻辑推理。