Here are the notes on logistic details such as picking up Exam 5 and the final exam. My schedule for the next few days is
| Thursday, May 6 | 8:00-10:00 am | Available for appointments |
| 11:00-noon | Office hour | |
| 1:00-2:00 pm | Available for appointments | |
| Friday, May 7 | 8:00-noon | Available for appointments |
| 3:00-4:00 pm | Office hour | |
| Monday, May 10 | 10:30-11:30 am | Available for appointments |
| 2:30-3:30 pm | Office hour | |
| 3:30-4:30 pm | Available for appointments | |
| Tuesday, May 11 | 8:00-11:00 am | Available for appointments |
Here is a list of objectives for the final exam. The list is a compilation of the objective lists for the individual exams.
In class, I mentioned some books you might want to look at if you are interested in learning more about vector calculus. Here's more information:
| Section | Problems to do | Submit | Due date |
|---|---|---|---|
| 9.1 | #1-45 odd | #14,42 | Monday, Jan 26 |
| 9.2 | #1-7 odd, 17-27 odd, 35-47 odd | #44,48 | Tuesday, January 27 |
| 9.3 | #1-19 odd,23,29,37,51 | #16,36 | Wednesday, January 28 |
| 9.4 | #5,9,15,17,21,25,27,29,31,37,45 | #36,38 | Monday, February 2 |
| 9.6 | #11,15,19,21,23,43,51 | None | |
| 9.5 | #17,19,21,25,27,29,31,33,47,49 | #24,50 | Wednesday, February 4 |
| 9.6 | #19,21,29,33,41,47,55 | #44,52 | Friday, February 6 |
| 9.2 | #11,15,29,31,33 | None | |
| 9.7 | #3-12,13,15,17,27,31 | None | |
| 9.5 | #3,5,7,9,11 | None | |
| 10.1 | #1,9,11,13,17,21,23,27,35,41,45,49,59 | #20,44 | Monday, February 16 |
| 10.2 | #1,3,7,9,15,17,19,25,27,31,35,51,59 | #38,50 | Tuesday, February 17 |
| 10.3 | #15,17,29,37 | None | |
| 11.1 | #1,5,7,11,13,17,35-40,43,47,51,53 | #14,50 | Monday, February 23 |
| 11.2 | #5,15,31,37,39 | None | |
| 11.3 | #3-21 odd,25,37,47,50 | #8,20 | Wednesday, February 25 |
| 11.4 | #1,7,13,17,23,25,29,31 | None | |
| 11.5 | #5,7,9,11,13,17,34,35,37,43 | #38,56 | Monday, March 8 |
| 11.6 | #1-15 odd, 27-33 odd, 41,43,45,49 | #34,40 | Wednesday, March 10 |
| 11.7 | #5,7,11,15,2125,27,29,35,41,57 | #26,44 | Wednesday, March 24 |
| 11.8 | #3,11,13,21,29,51,53 | None | |
| 12.1 | #13-19 odd, 23,27,37,47 | #26 | Tuesday, April 6 |
| 12.2 | #3,7,13,21,25,27,31,37,49,53,55,57 | #54 | Tuesday, April 6 |
| 6.3 | #3,11,19,23,27 | None | |
| 12.3 | #1,5,11,13,17,19,27,33,37,39,45,49,55 | #30,56 | Friday, April 9 |
| 12.5 | #5,7,11,17,19,27,31,41,47 | #32,48 | Monday, April 12 |
| 12.7 | #3-27 odd, 39,43(a),49,50,51,56,62,63(a) | None | |
| 13.2 | #7,9,13,17,19,25,27,29,31 | #14,26 | Friday, April 23 |
| 13.3 | #5-17 odd | None | |
| 13.5 | #23,25,27,29 | None | |
| 13.1 | #3,9,13,17,19,27,35,47,51,54,55 | None | |
| 13.6 | #3,7,17 | None | |
| 13.7 | #3,5,7,13 | None | |
| Extra problems | #1,2,3 | None |
You will need a PDF viewer to read the files posted here. Visit the Adobe website to obtain a free reader (all major platforms are supported).
You can look at exams from last time I taught Math 221. You can use these to get some idea of how I write exams. Don't assume I am going to write our exams by just making small changes to the old exams. You should also note that we were using a different textbook so some of the notation is different. There are also differences in the material covered on each exam. You can use this old exam to prepare for our exam in whatever way you want. I will not provide solutions for the old exam problems.
You can go to http://math.la.asu.edu/~kawski/vfa2/ to play with the Vector Field Analyzer written by Matthias Kawski of the Mathematics Department at Arizona State University. This Java Applet allows you to plot and analyze a two-dimensional vector field. Be patient as the web loads; it takes a minute or two for the Java Applet code to set everything up. After the code is loaded, you will see a plot of the default vector field. You may want to change to some simple example such as F(x,y)=(x,y). Note that the applet has three "layers" that you can change by clicking on the tabs labeled "Zoom lenses", "DEs/flows", and "Line int's." The "DEs/flows" tab is probably the one you want to start with.
I have assigned a lot of problems from Section 9.1 because there are a lot of small ideas to learn about vectors. In a week or two, all of these ideas will seem very simple.
Some of the problems are stated in a notation we did not get to today. This notation uses i and j where i is a vector of length 1 parallel to the x-axis and j is a vector of length 1 parallel to the y-axis. So, for example, the vector 3i-4j is the same as <3,-4>. You can use either notation.
After each in-class activity such as the one we did today, you will write a report. Your report should include a summary of results you got in class with your group and any results you got finishing up the activity outside of class. Your report should also include a discussion of things you learned from the activity and any questions you have. In writing this part of the report, you should ask yourself ``What are the main ideas in this activity?'' For today's activity (Which Way is North?), this discussion could reasonably be done in a paragraph. The report on this activity is due in class on Tuesday, January 27.
On Friday, we will first address questions you bring from the Section 9.1 homework and then look at new material.
Section 9.2 covers the same ideas as Section 9.1 but does so for vectors in space rather than vectors in the plane. For now, you can skip over the subsection entitled "Graphs in R3: Spheres and Cylinders"
When you compare how we did the dot product in class and how the text introduces the dot product, you will notice some differences. In class, I started with the geometric expression u•v=|u||v|cos&theta as a definition and then got the component expression u•v =u1v1+u2v2 +u3v3 as a result. The book starts with the component expression as definition and then gets the geometric expression as a result. Last Friday, I showed you how to go from the geometric expression to the component expression. To do this, we had to assume that the distributive property holds. I didn't show you an geometric argument for the distributive property. As an exercise, you should do this. That is, draw a picture to show that u•(v+w)=u•v+u•w. In doing this, think about the dot product as (component of first vector in direction of second vector) times (component of second vector in direction of second vector).
In Section 9.3, you can read the subsection "Direction Cosines" quickly. I am not putting any emphasis on this material and haven't assigned problems relevant to it.
In Section 9.4 of the text, the authors give a "determinant form" of the cross product. If you have seen determinants of matrices before, this may be helpful. Otherwise, ignore it. I won't use this in class. When I go to calculate a cross product, I write the vectors out in component form, one below the other with components lined up in columns. I then do the calculations by covering up the relevant column and working with the four remaining components in the pattern we did in class.
The last subsection is on "The Scalar Triple Product." We did not talk about this in class. For a few of the problems, you will need to use the result in the box on page 603.
We'll be covering planes in space from Section 9.6 before we talk about lines in space from Section 9.5. As you read Section 9.6, you should skip results and examples that involve lines in space. I've assigned some problems from Section 9.6 and I'll assign more after we have gone over the ideas from Section 9.5.
In Section 9.5, go straight to the subsection "Lines in R3" on page 610. We will discuss material from the first few subsections later.
The material we looked at today is in Section 9.2 (spheres and cylinders) and Section 9.7 (other quadric surfaces).
We will have an exam covering Chapter 9 next Tuesday, February 10. The exam will be from 11:00 to 12:20 am. You need to let me know as soon as possible if you are not available for the extended period of time on this day.
In preparing for the exam, you should make sure you have done and understood all of the assigned homework problems. Come talk with me or get help from a tutor if you have questions on any of these problems. You should also read the relevant sections of the textbook and review class notes. As you do so, make a list of questions and get answers for them. It is useful to write down your questions. Trying to phrase a question often helps you pinpoint exactly what it is that you are working to understand. If you have time, you can do some of the review problems at the end of each chapter. This can be useful because these problems are not in the context of a specific section and so are more like an exam situation.
Here is a copy of an old first exam from the last time I taught Math 221. You can use this to get some idea of how I write exams. Don't assume I am going to write our exam by just making small changes to the old exam. You should also note that we were using a different textbook so some of the notation is different and there are differences in the material covered by the exam. For example, the old exam covered vector-valued functions (which we have not done yet) and did not cover quadric surfaces (which we have done). You can use this old exam to prepare for our exam in whatever way you want. I will not provide solutions for the old exam problems.
You can use your calculator to plot the curve given by a vector-output function in the plane (i.e., a curve given by parametric equations x(t) and y(t) in the language of Section 9.5). I'll describe how to do this on a TI-86. The TI-83 is similar. I'm not sure about the TI-89. It may be possible to do space curves on the TI-89. On a TI-86, go to the MODE menu. One of the lines read Func Pol Param DifEq. Select Param and then go to the GRAPH menu. The F1 key is now labeled E(t)=. Hit F1 and the screen will show xt1= and yt1=. This is where you enter the x and y components of the parametric equations. Try x(t)=cos t and y(t)=sin t. Note that t is available on the F1 key. After you enter the component functions, exit the E(t)= menu and hit F5 to graph. You'll see the curve traced out and it will be, well, probably small and ovalish. Of course, it should be a circle but it will probably look like an ellipse because the standard window has different length scales along the horizontal and vertical axes. The best way to get equal length scales is using the ZSQR (for "Zoom Square") under the ZOOM menu. You can use the WIND menu to change the range of values for t. The default is to plot from t=0 to t=2π in steps of π/24.
You can get more details in your calculator manual. If you don't have a manual, you can download one on the TI web site: http://education.ti.com/us/global/guides.htmlFor the activity Acceleration, you might want to change the notation in 1(b) from r to R for the parametric curve. This would make the notation consistent with what the textbook uses and would prevent any confusion with the way r is used in the Warmup.
In class, I gave the following tentative dates for exams: Exam #2: Tuesday, March 2; Exam #3: Tuesday, March 30; Exam #4: ?; Exam #5: Tuesday, May 4 (in the last week of classes).
As part of your homework this weekend, you should do the problems on the handout I distributed at the end of class.
If you want your scores included in the web version of the gradebook, send me an e-mail with a code name. If you want to see what this will look like, go to the web page for some of my recent courses.
We will have our second exam on Tuesday, March 2 from 11:00 to 12:20. The exam will cover sections 10.1 to 10.3 and 11.1 to 11.4. For this exam, a well-prepared student should be able to
In class today, we had our first look at the gradient of a function of two variables. At each point in the plane, the gradient of a function is a vector. We first saw the gradient as it is given in cartesian coordinates. This is the expression with (partial derivative of f with respect to x) as the "i-hat" component and (partial derivative of f with respect to y) as the "j-hat" component. This is in the text at the bottom of page 739. This is all you need for Problems 1-9 of Section 11.6. The real question of interest is how do we define or understand the gradient vector geometrically.
Section 11.6 of the text starts with the topic of directional derivatives and then gets to gradients. In class, we will turn this around. In the long run, the idea of gradient is more fundamental that the idea of directional derivative.
Yesterday, I showed you some plots of gradient fields but didn't have copies because the copy machine was broken. You can look at these plots here. For higher quality graphics, you can download a PDF file.
Today, we did an activity entitled The Hill in class. The report for this will be due on Tuesday, March 23 (after Spring Break). If you want to use the level curve plots I made, you can download the PDF file here. If you missed class today (as an unusually large number of people did), you might want to find a group of others who missed class (i.e., other slackers) and work through the activity. You should use one of the first four points (A, B, C, or D but not E or F)
In class, we did not have time to do an example using the second derivative test for classifying (as local minimizer, local maximizer, or neither) a critical input for a function of two variables. The test is straightforward to apply. You should read about it in the text on pages 751 to 754 and try it out in the problems I have assigned from Section 11.7. That is, you should figure out how to use the test. We'll talk in class about why the test works. The text does not have any details on this.
For future activity reports, I will have higher expectations on completeness and style. In general, you should approach writing each report as if you were writing a solution manual for these activities. Assume the audience to be students in Math 221 who have covered the same material we have covered in class but have not done the specific activity. For the results part of the report, you should give a complete solution with enough detail for a reader to reconstruct your reasoning and calculations. For the "lessons learned" part, you should discuss what you view are the main ideas to be learned or reinforced in this activity.
In part, I am having you write these reports in order to practise technical writing. Here are some notes on technical writing in mathematics that might help with some of the style issues that arise.
I will be out of town for the first part of next week. I will be back and will check e-mail Thursday and Friday. You can send an e-mail if you have any questions. Have a good Spring Break.
I've made a change to this course web page so that you can see the scores I have recorded for you on assignments and exams along with a course total based on the weights given in the syllabus. In the contents list above, you will see the item "Check your scores." This link will take you to a form where you input your last name and UPS student ID number. If all goes correctly, when you submit this, you should get back a table with your score, the possible score, and the class average for each assignment and exam. If something goes wrong when you try this, send me an e-mail letting me know.
We will have our third exam next Tuesday, March 30 from 11:00 am to 12:20 pm. This will cover Sections 11.5 through 11.8 and possibly the first part of Chapter 12.
In doing the problems from Section 11.8, you should use the method of Lagrange multipliers even if it looks easier to solve the constraint for one variable and substitute into the objective to get a "Calc I" problem.
We will have our third exam next Tuesday, March 30 from 11:00 am to 12:20 pm. This will cover Sections 11.5, 11.6, 11.7, and 11.8. I will post a list of objectives for this exam by the end of the week. To prepare for the exam, you should read the relevant sections of the text and work on the assigned problems (both those to be submitted and those not submitted). Keep a list of questions that come up in your reading and work on the problems. Get answers to these questions from the many sources available to you:
In the last part of class today, we started working through an argument for why the second derivative test works. We did not finish the argument. You can read the remainder of the argument in this handout. I will not hold you responsible on an exam for understanding this argument. I will expect you to be able to use the second derivative test.
Exam #3 will cover Sections 11.5, 11.6, 11.7, and 11.8. In Section 11.7, you can skip the subsection entitled "Least squares appoximation of data." For this exam, a well-prepared student should be able to
Today, we defined two things: double (definite) integral and iterated integral. Fubini's theorem tells us that these things, though defined differently, are equal. In the geometric view, for the same function and domain, a double integral and each iterated integral give us the same volume. The double integral corresponds to chopping a loaf of bread into pieces with a rectangular cookie cutter. Each iterated integral corresponds to slicing the loaf of bread in the usual way one would cut up a loaf. In general, we will use Fubini's theorem to trade a double integral in for an iterated integral because we can often evaluate the iterated integral using the Fundamental Theorem of Calculus (twice).
I have assigned a lot of problems from Sections 12.1 and 12.2. Often the hardest part of a problem, particularly in Section 12.2, is to correctly describe the region of integration with appropriate upper and lower bounds for x and y. This requires good visualization and drawing skills. After you have set up the correct interated integral, evaluating it should be a straightforward process. You will need to recall how to find antiderivatives. Remember techniques such as substitution and integration by parts. It's also fair to use the table of integrals in your text or one in some other resource.
You should read about polar coordinates in the first part of Section 6.3 on pages 375-379 of the text. It will be important to learn about polar graphs. Note that you can produce a polar graph on most graphing calculators. For example, on the TI-86, go to the MODE menu where you can choose the option Pol on the fifth line. If you do so and then go to the GRAPH feature, you will see that it is set up for a polar function r(&theta). Work on the problems I have assigned from this section. On Monday, we will use polar coordinates to evaluate double integrals for regions that are most easily described in polar coordinates.
I have assigned a due date of Tuesday, April 6 for the Problems from Sections 12.1 and 12.2.
Many of the problems with double and triple integrals involve being able to visualize and draw regions of space. This can be very hard. You should try to draw a perspective view of the three dimensional region. In cases with a lot of symmetry (as most of the problems have), it is useful to look at a cross-section. This is easier to draw because the cross-section is a planar region. In many cases, looking at a cross-section (or two) will help you see how to describe the given region with appropriate lower and upper bounds on the coordinates.
We are skipping Section 12.4 for now. We will skip Section 12.6 altogether. If you have taken or are in physics, you might want to look at Section 12.6. This section is about calculating physical quantities such as center of mass and moments of intertia. Physicists nowadays often use the phrase rotational inertia rather than moment of inertia.
In Section 12.7, we have only talked about cylindrical coordinates so far. For the problems from this section, just do the ones relevant to cylindrical coordinates and not spherical coordinates.
In trading in a triple integral of a function f over a region D of space for an iterated integral in some chosen coordinate system, you need to take care of three things:
Remember that there are different conventions for the notation and ordering of spherical coordinates. I will use (ρ, φ, θ) where φ is the angle between the positive z-axis and the ray from the origin through the point. The text (and most math texts) will use (ρ, θ, φ) with φ having the same meaning. Most physics texts will use (r, θ, φ) with θ is the angle between the positive z-axis and the ray from the origin through the point. You can read a little more about this issue at http://www.physics.oregonstate.edu/bridge/papers/
We will have Exam #4 on Friday, April 16 from 11:00 to 11:50. This will cover the material from Sections 12.1, 12.2, 12.3, 12.5, and 12.7.
Exam #4 will cover Sections 12.1, 12.2, 12.3, 12.5, and 12.7. For this exam, a well-prepared student should be able to
Today, we began our last big topic of the course, namely the calculus of vector fields. This topic is often called vector analysis. I will be doing vector analysis in an order and style that is not the same as the text. I will try to keep you updated on the relevant sections of the text and on being comfortable with both approaches.
As part of your assignment for today, you should check out the Vector Field Analyzer (written by Matthias Kawski of the Mathematics Department at Arizona State University ). This Java applet allows you to plot and analyze a two-dimensional vector field. Be patient as the page loads; it takes a minute or two for the Java code to set everything up. After the code is loaded, you will see a plot of the default vector field. To plot a different vector field, enter the components of the vector field you want in the boxes near the bottom of the applet window and then click on the button "Plot this field." You should try the simple examples we did in class and the examples given in the first part of Section 13.1 in the text. The Vector Field Analyzer has lots of features (and a few bugs). We'll explore more of the features in connection with different parts of vector analysis. If you want to play right now, click on the tab labeled "DEs/flows." This part of the program lets you draw a box on the vector field plot and then watch as the box "goes with the flow." Under the tab "Line int's" you can draw a curve on the vector field plot and see the value of the line integral. (Click off the check box labeled "Show flux" first.) The value of the line integral is given as "Circ" which is an abbreviation for circulation. We will talk in class about why the line integral is labeled as circulation here.
In the text, you should read the first two pages of Section 13.1. This is the subsection entitled "Definition of a Vector Field." You should then read the second subsection of Section 13.2 entitled "Line Integrals of Vector Fields." The definition of line integral given in the box on the bottom of page 870 uses a parametric description of the curve C. In class, I am going to emphasize a non-parametric approach to line integrals. I have written a handout that gives an example of both approaches so you can compare. (I haven't had time to proofread carefully or add pictures to this handout. Let me know if you spot any mistakes.) You can look at the homework problems but don't worry too much about these yet. The handout and the activity we do tomorrow in class will help you understand how to compute line integrals. We'll answer questions on these problems in class on Wednesday and the submitted problems are not due until Friday.
You should finish the activity we started in class. After you complete Parts 3 and 4 for the curve your group was assigned, you should repeat these for curve I. We will talk about some of the ideas from this activity in class tomorrow. We will also look at any of the homework problems from Section 13.2 on which you have questions.
The activity from Tuesday leads us directly to the Fundamental Theorem for Line Integrals. In the text, this is given in Section 13.3. You should read all of section 13.3 with the exception of Theorem 13.4 and Example 5. This theorem and example deal with using the curl of a vector field in space; we have not yet talked about curl so we will come back to this later. Note that Theorem 13.3 gives a test that you can use to determine if a given planar vector field has a potential function or not. Example 3 illustrates how you can find a potential function for a conservative planar vector field. I have written a handout that shows a slightly different way to find a potential function. You should work on the problems I have assigned for Section 13.3. These deal only with planar vector fields for now.
For Tuesday's activity, you should turn in solutions for the various for both curves (the one your group did and curve I). You should make nice pictures for these and give complete, neat solutions. You do not need to write a formal paper with complete sentences and full explanations. You should include some brief discussion of the lessons learned or reinforced by the activity. This is due Monday, April 26.
I have updated the two handouts from this week. "Notes on line integrals" now includes a few figures. "Finding potential functions" has an additional example and a new section on thinking geometrically about potential functions. This new section explains how to use some tools in the Vector Field Analyzer for this geometric thinking about potential functions.
In class today, we looked at surface integrals. These are covered in Section 13.5 of the text but in what will look like a very different way. The first part of Section 13.5 defines surface integral for a scalar function of three variables. The second part entitled "Flux Integrals" is more closely related to what we did in class. The style used for computing surface integrals is very different. You might want to skip reading this part of the text for now and focus on trying the assigned problems with the approach we did in class. Here's the handout "Notes on surface integrals" with some details on a definition of surface integral and an example of computing a surface integral. (Note: I made some changes to this on Saturday, April 24 with some minor wording changes and a few more figures.) I recommend you look at the figures on your screen since these might not print well. On screen, you can zoom in to see more detail.
In class today, I did a miserable job of drawing pictures for Problem 23 from Section 13.5. Here's an attempt to make up for this. (Note: I have moved the surface integral plots to a separate page so this main page can be downloaded quickly.)
For a vector field, there are two useful notions of derivative, namely divergence and curl. You should first focus on becoming proficient in computing divergence and curl. You should also begin building intuition for thinking about what divergence and curl tells geometrically about a vector field. There are features of the Vector Field Analyzer that can help with this. On the "DEs/flows" tab, you can draw a box or circle in the plot window and watch it move under the flow (thinking of the vector field as giving velocity vectors for fluid flow). Look at the change in area and at the rotation as the box or circle moves along.
In the text, you should read Section 13.1. You can skip the subsection entitled "The Laplacian Operator." If you have had some physics, you might want to read the paragraphs starting at the bottom of page 865 through the end of the section. The notation used here may differ from your physics text but you should be able to make some connections.
I handed out "A map of vector calculus" as a tool to help you see the many connections within vector calculus. Today, we saw how line integrals, surface integrals, divergence, and curl come together in Stokes' Theorem and the Divergence Theorem. You should read Sections 13.6 and 13.7. Now that we have talked about curl, you should also go back and read Theorem 13.4 and Example 5 in Section 13.3. (This concerns the connection between curl and conservative vector fields shown as a dashed arrow on the map of vector calculus.) On Friday, we should have time in class to talk about the proof of Theorem 13.4 that is given on page 912. In Section 13.6, you can skip Example 4 unless you are interested in the physics of that example. In Section 13.7, you can skip the subsection "Applications of the Divergence Theorem" or just read those examples that catch you eye if you are interested in physics.
Exam #5 will cover Sections 13.1 through 13.7 and the material on the three handouts
Check previous "Daily notes" for what parts of the text to emphasize and what parts to skip.For this exam, a well-prepared student should be able to
In class today, we looked at an informal proof of Stokes' Theorem. I am writing a handout with the details and will put it here when I finish later this weekend. You should work on the assigned problems from Section 13.6 and 13.7 and from the handout "Some problems on Stokes' Theorem" I distributed in class today.
You can use the Vector Field Analyzer to explore the ideas of circulation density and flux density. In the applet, select the "Line ints" tab. Click the check box for the option "Show circ" box. Make sure the default "Box" option is selected. Click and drag in the plot window to draw a box. The box will appear with colored regions along its edges. These indicate the contribution to the circulation (that is, to the line integral around the box). Green indicates positive contributions to the circulation and yellow indicates negative contributions. The width of the colored region indicates the magnitude of the contribution. The sum of these contributions is the circulation and its value is shown in the region just below the tabs. This region also shows the area enclosed by the box and the ratio of circulation to area. This ratio is what we take the limit of to get the circulation density. Now select the option "Resize curve." With this selected, you can drag in the plot window to make the box bigger or smaller. Watch what happens to the value of the ratio as you make the box smaller. You should see this approach a constant value. This is the limit of the ratio and hence the circulation density.
You can play a similar game with the "Show flux" option checked. A box drawn in the plot field will now have pink and blue regions along the edges. These indicate contributions to the flux. Blue indicates negative contributions to the flux and pink indicates positive contributions. The sum of these contributions and the ratio of this sum to the area are given in the region just below the tabs. By selecting the "Resize curve" option, you can look at these values in the limit as the box gets small. This limit is the flux density.
I suggest you experiment with some very simple vector fields such as F=x i+y j and F=-y i+ x j to begin with. Compute divergence and curl for these vectors fields and compare the results with the flux density and circulation density you get using the Vector Field Analzyer.
One last note: the flux density we discussed in class is for a vector field in space and thus is flux per unit volume. The flux density in the Vector Field Analyzer is for a planar vector field and thus is flux per unit area.
Here is the handout "Circulation density and an argument for Stokes' Theorem" that covers what we did in class on Friay. I recommend you look at the figures on screen so you can zoom in to see detail. Some of labels are a bit small.
Here's what I want you to get out of material: