Bridging the Vector Calculus Gap

在数学家通常教授向量微积分的方式和其他科学家使用向量微积分的方式之间存在“向量微积分差距”。这种材料对物理学家和许多工程师来说是必不可少的，因为它在描述电和磁方面起着核心作用。这个项目的两个基本基础是使用几何推理而不是算法计算（讲课的新重点）和使用开放式小组活动（背诵的新重点）。我们认为，迄今为止，我们的主要成功是识别几何推理，使用向量微分，作为所有向量微积分的共同主题。在这个项目的前一个（概念验证）阶段，我们基于这种方法开发了小组活动，一些用于向量微积分课程，一些用于高年级物理课程的相关材料。我们和几个机构的其他人已经成功地使用了这些活动。这个（全面发展）项目将通过向其他教师提供使用这些材料的材料和培训来“锁定”我们的成功。已开发的教师指南包含有关矢量微积分的几何方法的信息，有效使用小组活动的建议，以及关于个人活动的提示。增强学生对矢量微积分的几何理解，有助于弥合“矢量微积分鸿沟”。

There is a "vector calculus gap" between the way vector calculus is usually taught by mathematicians and the way it is used by other scientists. This material is essential for physicists and many engineers due to its central role in the description of electricity and magnetism. The two basic underpinnings of this project are the use of geometric reasoning rather than algorithmic computation (a new emphasis for lectures) and the use of open-ended small group activities (a new emphasis for recitations). We believe that our major success so far has been the identification of geometric reasoning, using the vector differential, as the common theme underlying all of vector calculus. In the previous (Proof of Concept) phase of this project, we developed small group activities based on this approach, some intended for use in a vector calculus course, and some for use in upper division physics courses on related material. These activities have been used successfully by us and by others at several institutions. This (Full Development) project will "bottle" our success by providing materials and training in the use of these materials to other faculty. The developed Instructor's Guides contain information about this geometric approach to vector calculus, advice on using small group activities effectively, and tips on the individual activities. Enhancing students' geometric understanding of vector calculus will help to bridge the "vector calculus gap".