We talked about 6.15 and 6.16, especially about 6.16 part 2. We went on to 6.18 and talked about how partial derivatives can tell us directions of movement. Then we reviewed the chain rule from calculus 1 and did 6.24, which generalizes the chain rule to multivariable calculus. We skipped around a bit to make sure that we introduced the subjects you need to do the rest of the problems in the chapter.
We won't have write-ups due on Monday. In order to cover the material we need to cover this semester, our goal is to finish the chapter on Monday. Do at least 6.19 and 6.25-6.29 to prepare for Monday, and try to finish the rest of the problems (6.17-6.33).
We went through 6.19, 6.25-6.27, and 6.29-6.30. We talked about the big ideas of how the chain rule helps us calculate how changes in the inputs affect the outputs in complicated situations, and spent a lot of time talking about how to view things graphically.
Writeups due on Wednesday are: 6.19, 6.25, 6.27, 6.29, and 6.31.
Now that we've finished talking about how derivatives generalize to multiple-variable situations, next time we will start discussing how integrals generalize.
We went through problems 8.1, 8.2 (setup), 8.3.1, and 8.5.1. We reviewed the big idea behind integrals and generalized those to the concept of line integrals, and talked about how a line integral can calculate surface area, curve length, as well as the mass of a wire (we'll study the mass problem more later).
To prepare for next time, do all of 8.1-8.5 (including computing the integral in 8.2). You'll be turning in 8.1, 8.2, 8.3.1, and 8.5.1. Also, think at least a little bit about the challenge problem after 8.5.
We'll also move on and talk about the work problems starting in section 8.2. Please prepare for this by reading through problem 8.7, watching the video in the margin by problem 8.7, and trying 8.6 and 8.7.
In the video, it talks about $d\vec r=\vec T ds$. $\vec T$ is just the unit tangent vector $\vec r'(t)/|\vec r'(t)|$. $\vec T ds$ is just breaking $d\vec r$ up into its direction (as a unit vector, $\vec T$) and its length $ds$.
We went through the problems or parts of problems in 8.1-8.5 that we didn't cover last time, and we talked about the challenge problem as well. Then we spent some time doing problem 8.6 and talking about calculating work (the discussion is summarized in the text.)
For Monday, please prepare the problems in section 8.2. Please watch the Youtube videos linked in the margin and use Sage to check your work. We won't be turning anything in on Monday, but please come prepared to finish talking about section 8.2.
We went through problems 8.7-8.10. For Wednesday, please prepare through problem 8.16. I highly encourage you to watch the Youtube videos Ben has made available via the links in the margins. You won't need to turn in problems on Wednesday—use your several hours of studying to prepare what we will talking about in class or reviewing what we've covered.
We went through problems 8.11-8.16, and talked about how to view flow and flux geometrically as an area of fluid that flows along or past a curve. We talked a lot about how the choice of normal vector $\vec n$ ($(dy,-dx)$ or $(-dy,dx)$), the direction of motion along the curve, and the sign of the flux are all related. We also practiced quite a bit with looking at pictures of situations (curves on vector fields) and determining if the flow or flux was positive or negative.
Originally I said we wouldn't be having class on Friday, but my flight (and therefore trip) was canceled due to weather, so now I *will* be there in class on Friday to help you through the problems in section 8.3 and 8.4, as well as help introduce problems 8.27-8.29. I think it will be very helpful to come. Please come having worked as far through the problems in sections 8.3 and 8.4 as you can (working through problems 8.17-8.22 would be reasonable).
For Monday, please prepare through (and including) problem 8.29.
A couple of us met and talked about the averaging problems in sections 8.3 and 8.4.
We did some problems that reviewed the major ideas in sections 8.3 and 8.4, and introduced section 8.5. For Wednesday, please watch the Youtube videos and attempt the remaining problems in the section.
We will have our second test next Wednesday, covering chapters 5, 6, and 8.
We did discussed problems 8.27-8.31, and we also talked about 8.33.
For Friday, please turn in write-ups for problems 8.29-8.31. For 8.29, follow the hint in square brackets and explain in sentences what is going on. On Friday, we will be starting the next unit, chapter 9, on optimization. I updated the online problem notes to include chapter 9.
We discussed the material in problems 9.1-9.12. Please make sure that you can do all of these problems (we did many of them in class). The textbook also talks about this material in textbook sections 13.6 and 13.7 (though the textbook doesn't talk about the extensions to higher dimensions that we can talk about because we understand the total matrix derivative).
For Monday, please turn in write-ups for problems 9.5, 9.6, and 9.10. On Monday, we will have some time for questions on the test (remember: test 2, covering chapters 5, 6, and 8, is on Wednesday, 1 May). We will also cover section 9.2 on Monday.
We answered questions about the test and discussed section 9.2.
We took our second exam over chapters 5, 6, and 8.
We reviewed section 9.2 and discussed section 9.3.
For Monday, please turn in problems 9.16 and 9.18. On Monday, we will discuss double and triple integration, and we will have a little time for questions over Lagrange multipliers (section 9.3).
You can also see more examples of the second derivative test and Lagrange multipliers in the last two sections of chapter 13 in the textbook. For the second derivative test, the textbook uses a special case of the method we discussed in class—our problem 9.18 discusses the correspondence between the book's method and our more general method.
I posted the worksheet we looked at in class here: http://sage.cs.drake.edu/home/pub/71/
We discussed double integrals, describing regions with inequalities, finding volumes and averages (including center of mass, centroid, etc.) using double integrals. We also briefly discussed how to do substitution with multiple variables.
For Wednesday, please do 10.5, 10.14, and 10.16. Sections 14.1-14.4 in the textbook also cover this material.