Vector Calculus via Linearization: Visualization and ModernApplications

该项目正在开发一种全新的矢量微积分方法，该方法与最近其他大学入门级数学、科学和工程 (MSE) 课程的改革努力完全一致。作为这个更大的 MSE 课程的一部分，它带来了更多的连贯性和一致性，强调了大部分课程的共同基础。复杂的计算机可视化得到了大力利用，因为它为矢量微积分开辟了一条全新的途径，允许将矢量导数的概念重新连接回其基础局部线性。学生现在可以通过缩放看到卷曲。与线性代数和微分方程概念的联系无处不在。该课程非常适合合作学习环境，引导学生通过经常戏剧性的计算机实验自己做出许多关键发现。新课程还通过利用具有普遍应用的现代几何控制来扩大应用范围（超越传统的电磁和流体动力学），甚至与适合这门大二课程的最新研究成果相联系。课程以互动文本的形式实施，并利用现代电子媒体和WWW进行快速传播。对增强学习体验的评估是该项目的一个组成部分。该课程通过传达连贯且现代的高等数学观点，对 MSE 专业以及未来的数学和科学教师产生重大影响。

This project is developing a radically new approach to vector calculus that is completely aligned with recent reform efforts in other college entry-level math, science, and engineering (MSE) courses. As part of this larger MSE curriculum, it brings more coherence and consistency to it, emphasizing the common foundation for much of these courses. Sophisticated computer visualization is vigorously exploited as it has opened a completely new avenue to vector calculus, allowing to reconnect the concepts of vectorial derivatives back to their foundation, local linearity. Students now can see the curl by zooming. Connections to concepts of linear algebra and differential equations perspectives are omnipresent. The curriculum is ideally suited for cooperative learning environments where students are guided to make many key discoveries themselves via often dramatic computer experiments. The new course is also broadening the range of applications (beyond the traditional EM and fluid dynamics) by drawing on modern geometric control which has ubiquitous applications, even connecting to recent research achievements that fit into this sophomore course. The curriculum is being implemented in the form of interactive texts, and is using modern electronic media and the WWW for rapid dissemination. Assessment of the enhanced learning experiences is an integral part of the project. This curriculum is having a high impact on both MSE majors and future math and science teachers by conveying a coherent and modern view of advanced mathematics.