Lesson 15: Scalar Surface Integrals; Vector Fields

Preview

In the previous lesson we learned how to calculate surface area. In this lesson we’ll extend that notion to integrate a function over a surface, yielding scalar surface integrals. We'll also introduce the notion of a vector field and explore a few important types of vector fields.

  • In Module 1 we’ll define what a scalar surface integral is and discuss how it contains the surface area formula from the previous lesson as a special case. We’ll also compare scalar surface integrals to line integrals and discuss their parallels.

  • In Module 2 we'll introduce the notion of a vector field, learn how to plot vector fields, and discuss what those plots tell us about the vector field.

  • Finally, in Module 3 we'll go on a brief tour of important types of vector fields, including gradient fields and the divergence and curl of a vector field.

Learn

Work through the lesson notes below, consulting the videos below it when you get to the "See Class Notes" boxes. For your records, the annotated lesson notes are below the videos. Some tips for you as you work through these resources:

  • I recommend using Cornell Notes (or a modification of it; see this video starting at the 1:05 mark) to take notes on the lesson and the videos. This note-taking method balances detail with big-picture thinking to help you summarize and retain what you are learning. See this other video for additional note-taking techniques you might want to experiment with.

Lesson Notes

Lesson 15.pdf

Module 1 Video

Module 2 Video

Module 3 Video

Reflect

If you are currently enrolled in this course with me, submit the written reflections Google Form I have emailed you after working through the lesson notes and videos. Some tips:

  • Submit substantive, but concise, answers to each question; you will be doing the future you a big favor by taking time now to accurately and succinctly summarize what you have learned from the lesson.

  • Send yourself a copy of your reflections; they will come in handy later when you start preparing for quizzes and other assessments.

If you are not currently enrolled in this course with me, those written reflections ask three reflective questions designed to help you retain what you've learned and pinpoint any remaining areas of confusion. Those questions are:

  • Please summarize the main mathematical takeaways from the lesson notes.

  • What was the most interesting part of what you learned, and why?

  • What, if anything, do you still find confusing?

Practice

Work through the practice problems suggested below to see how much of this lesson you've understood.

Lesson 15 PP.pdf