Vector Fields

Sayonita Ghosh Hajra, California State University-Sacramento, Mathematics and Statistics

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Summary

This activity is exploratory and collaborative MATLAB activity. Students will explore and visualize vector fields. Students will also learn about the flow lines of the vector fields.

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Learning Goals

This class activity is intended for two 50-minute class periods. The learning objectives of this activity is to familiarize with vector fields, sketch vector fields, and draw flow lines. Particularly, students will
- learn how to sketch vector fields in 2D 
- explore the flow lines of vector fields
- develop writing skills to explain mathematics in their own words
Student will get to sketch some of the vectors and guess how the vector field will look like. Then they will use MATLAB to check their hand drawn vector fields and verify their guesses. Also, students will use the MATLAB sketches to make sense of flow lines of the vector fields.

Context for Use

California State University, Sacramento is a public university in Sacramento. This activity is intended for a multivariable calculus course. The course is a continuation of Calculus II for students majoring in mathematics, physical science and engineering. The class size is usually 30 and is offered 50 minutes for four days a week. This classroom activity is intended for two 50-minute class periods. For this activity, students are expected to be familiar with the MATLAB interface, and mathematical operations. This activity is for the 12th-13th week of a 15-week semester; this activity could engage students more productively in understanding and visualizing vector fields.

Description and Teaching Materials

This activity can be used as an introduction to vector fields. A short review of vectors is added to help students to sketch the vectors by hand first. This activity can be either conducted in a class with personal laptops or in a lab. Students can work in pairs for this activity. I chose MATLAB as it has a good collection of toolboxes, it is available for free to all students in my school and it is highly likely that students have used MATLAB in other courses as students are mainly engineering and computer science major.
This activity is planned for two 50-minute class period. In the first class, vectors are revisited and students are introduced to vector fields. First students sketch few vectors by hand to get a sense of the vector fields. Then they generate the vector fields in MATLAB to verify their hand-drawn sketches. On the first day of the activity, students complete class activity 1, examples 1-6. Students are also briefly introduced to flow lines towards the end of the class.
Students will be asked to complete the homework before next class meeting. First, they will be asked to sketch few vectors by hand and guess how the vector field will look like. Then they will check their answers by generating the vector fields using MATLAB. Students upload their work in the Learning Management System (Blackboard, Canvas, etc.) before next class meeting.
In the next class meeting, instructor will bring hard printouts of the student generated vector fields. Instructor will form groups of two or three. Students will work on sketching the flow lines of the vector fields. Each group will be given one of the sketches from the homework problems. Students will work on this activity for 15 minutes. Then groups will exchange their papers and review the other group's work. Next, students will present their work in a brief presentation. The rest of the class meeting will be spent on answering and clarifying student questions. Students will make corrections on the homework problems and resubmit their files if needed.

Grading Rubric (Acrobat (PDF) 48kB Aug31 20) 
MATLAB Activity (MATLAB Live Script 1.3MB Aug31 20) 

Teaching Notes and Tips

- Content Specific- This activity serves better after the discussion of how to plot vectors in two-dimensions. For example, a vector <1, 2> can be obtained by starting at (0,0) going forward 1 unit horizontally and 2 units upwards and then drawing the vector using an arrow from (0,0) to (1, 2). This process can be used to draw vectors from points other than the origin. Also, one can see if the signs of the components are both positive, then the vector points in the northeast direction. Therefore, I included a small section as review of vectors to help students sketch some random vectors at various points on the plane.
- Group organization- It is useful to form groups for this activity; groups of two is ideal but could be modified for three as well.

Assessment

This activity is assessed through a form of formative assessment. This aligns with the learning objectives for the class activity, which is to assess whether students (individually and collectively) are learning and provides an opportunity for the instructor to check students' understanding of prior knowledge of vectors.
Students will submit their MATLAB files in the LMS. This part of the activity will be graded for completion, using the attached rubric.

References and Resources

MathWorks. (n.d.). Quiver. https://www.mathworks.com/help/matlab/ref/quiver.html

Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning, Inc.

Rogawski, J. & Adams, C. (2015). Multivariable Calculus: Early Transcendentals (3rd ed.). W.H. Freeman & Company.