A Multivariable Calculus Manual to Accompany 3D Manipulatives

配合 3D 操作的多变量微积分手册

• 批准号: 0941877
• 负责人: Daniel McGee
• 金额: $ 6.67万
• 依托单位: University of Puerto Rico Mayaguez
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0941877&HistoricalAwards=false
• 关键词: Multivariable; Calculus; Manual; Accompany; 3D

Mathematical Sciences (21): This project, in two phases, is producing and testing an inexpensive kit, along with accompanying materials for its use, which allows students to effectively visualize points, vectors, lines, planes, and surfaces in 3-D. Many students currently visualize only 2D topics in their natural dimension and then must generalize to n dimensions. Mathematical software packages such as Mathematica, Maple, MathCAD, and Matlab have provided enormous aid to students and professors wishing to visualize concepts in three dimensions. However, there are many concepts where the two dimensional nature of a computer screen can limit the effectiveness of these packages, particularly if students have a weak geometrical background. For example, directional derivatives require the tangent line to a surface in a given direction associated with the xy plane. In three dimensions, a surface can be placed over the xy plane, the direction on the xy plane can be indicated, and the concept can be visualized quite easily. However, visualizing a precise direction and its associated tangent line on a 2D computer screen is often difficult for students. Correspondingly, a more effective pedagogical approach for this and many other concepts is to use physical 3D manipulatives. These allow visualization and motivation of concepts in a real three dimensional space. Manipulatives often prove more effective than a projection of three dimensions onto a two dimensional computer screen, particularly when students are first being introduced to multivariable functions. With this kit, students can visualize both 2D and 3D topics in their natural dimension before generalizing to n dimensions. The initial phase of this project has produced the basic set of manipulatives and they have been successfully tested. This new phase of the project is producing the materials which allows both students and teachers to use the manipulatives effectively.

数学科学（21）：这个项目分两个阶段，生产和测试一个便宜的工具包，沿着材料，使学生能够有效地可视化点，矢量，线，平面和表面的三维。目前，许多学生只在自然维度中可视化2D主题，然后必须推广到n维。数学软件包，如Mathematica，Maple，MathCAD和Matlab为希望在三维空间中可视化概念的学生和教授提供了巨大的帮助。然而，有许多概念，其中计算机屏幕的二维性质可以限制这些软件包的有效性，特别是如果学生有一个薄弱的几何背景。例如，方向导数要求在与xy平面相关联的给定方向上的曲面的切线。在三维空间中，一个表面可以放在xy平面上，xy平面上的方向可以指示，概念可以很容易地可视化。然而，在2D计算机屏幕上可视化精确的方向及其相关的切线对于学生来说通常是困难的。相应地，对于这个概念和许多其他概念，一个更有效的教学方法是使用物理3D操作。这些允许在真实的三维空间中的概念的可视化和激励。操纵往往证明更有效的三维投影到一个二维的计算机屏幕上，特别是当学生第一次被介绍给多变量函数。有了这个工具包，学生可以在推广到n维之前，在自然维度上可视化2D和3D主题。该项目的初始阶段已经产生了一套基本的操纵器，并已成功测试。这个项目的新阶段是制作材料，让学生和教师有效地使用操作。