If things are working correctly, you should see a picture below this text. There may be a brief delay while the Java applet code is set up. By dragging with your mouse, you can rotate each picture to see things from different views. You can zoom in and out by holding the shift key while you drag the mouse vertically within the plot region. The picture shows a parallelepiped and some of the geometry we talked about in class. This version is a bit crude. I may have an improved version later.
Some of the figures in the text show objects in space from a fixed viewpoint. It's nice to see to change the viewpoint to get a better sense of the object. I'm experimenting with providing this using Mathematica (a commercial product) to generate the plots and LiveGraphics3D (a Java applet written by Martin Kraus) for the display and control on the web page. Below are two of the plots from Figure 14.4 on page 481. By dragging with your mouse, you can rotate each picture to see things from different views. (In each, the surfaces are black when viewed from "underneath." I'll work on fixing this later.)
Below is a picture of the "spherical box" defined as the points with spherical coordinates ranging in intervals of extent dr, dφ, and dθ. From the geometry of this box, we can determine the correct volume element for spherical coordinates. You should finish off the handout we started in class on this. You should then test the result in a case for which you already know the correct result. A good test case is to compute the volume of a sphere using a triple integral (with integrand equal to 1) in spherical coordinates. This is very easy to do.
Here are solutions for the handout Some triple integral problems. I've written solutions for the first three problems since most of you got a correct solution for the fourth problem while I...well, I'd rather not say. Let me know if you have questions on these or spot any errors. I did not include any pictures in these. You should visualize and sketch the relevant pictures if you read through a solution.
Below is a plot showing the graph of the function f(x,y)=2xy/(x2+y2) for inputs near (0,0). The plot also shows the line segment from (0,0,-1) to (0,0,1) in red. This line segment is not part of the graph. (The line segment does not show up well from all viewing angles.)
Last spring, I wrote some notes on line integrals for my multivariate calculus course. You might want to have a quick look at these because they better reflect how I'm approaching line integrals now (as opposed to when I was working on the integrated calc/phys textbook.) Note that the reference to "the text" in these notes is a reference to the text used in the "non-integrated" calculus sections.
You might want to look at two more handouts from last spring, one on potential functions and the other on surface integrals. The handout on surface integrals has a better picture (in color!) than anything in the text. Below on the right is a version of Figure 1 you can rotate to view from different angles. On the left is a plot of the surface, area element vectors (in red), and vector field outputs (in blue) for the example. I've plotted the area element vectors on the same side of the surface as the vector field outputs. (Note: Somethings a little off with the magnitudes of the area element vectors. I'll fix this later.)
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 have looked at in class such as F=x i+y j and F=-y i+ x j . The Vector Field Analyzer has lots of features (and a few bugs). Here are some things you might want to explore now.
Suppose we have a planar vector field F. We want to think about whether F has a potential function or not. By definition, the function V is a potential function for F if ∇V=F. Recall that at a point P(x,y), the gradient vector ∇V(x,y) is perpendicular to the level curve of V that goes through P. If we have a plot of the given vector field F, we can start by drawing, at the base of each vector, a short line segment perpendicular to that vector. The question we have to ask is Can we connect these line segments to form level curves for the potential function V? The level curves of a function can not intersect. (You should think through why this is so.)
To test out this idea, draw a vector field plot for each of the vector fields F=x i+y j and F=-y i+ x j . On each vector field plot, draw the perpendicular line segments and see if you can connect these up to form sensible level curves.
The Vector Field Analyzer provides some tools to help with this geometric view of potential functions. The first is right above the ``Plot this field'' button. You will see the phrases ``Arrows (contra-var)'' and ``Stacks (co-var).'' The default is ``Arrows''. Click on the button closest to ``Stacks'' and look for the change in the plot window. Each arrow is replaced by a ``stack'' of line segments perpendicular to the corresponding arrow. The density of the stack is proportional to the length of the corresponding arrow. To make this a little more obvious, you might want to change the value in the box labeled ``grid'' just under the second component window near the bottom. The default value is 20 meaning that arrows/stacks are plotted on a 20 by 20 grid for a total of 400 arrows/stack. If you change this value to 10 (and then hit the ``Plot this field'' button), you get a 10 by 10 grid. With fewer arrows/stacks, each can be bigger without overlap. These stacks represent pieces of potential level curves. Can these be joined up into sensible level curves? Try the vector fields F=x i+y j and F=-y i+ x j .
The second tool is under the ``DEs/flows'' tab. On this tab, click on the button labeled ``Equipot. candidates.'' Then, click on one or more points inside the plot window. A blue dot will be drawn at each point on which click in the window. Finally, click on the button labeled ``Stop and Go.'' For each point you made, the program will start drawing a curve that is perpendicular to the vector field arrows (and parallel to the stacks if you are in that view). Experiment here with the same two vector fields used above.
We looked at the Vector Field Analyzer in class today. The note for Friday, April 1 describes some features of this applet that are relevant to the question of whether a vector field is conservative or not. In class, we looked at some features that are useful in building a geometric interpretation of divergence.
The first tool is the "Zoom lens" labeled "Div". With this selected, moving the cursor over the vector field plot will show a zoom with part of the vector field outputs subtracted to reveal the amount that the vector field points toward or away from the point at which the zoom is based. There are options to control the size of the lense window and to control the scale at which the vector field outputs are plotted. (Try moving the "range" slider to see this change.)
The second tool is on the tab labeled "Line ints". With this tab selected, check the option "Show flux" and then use the cursor to draw a box in the vector field window. The applet will display have pink and blue regions along the edges of the box. 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. 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.
I am developing the material in the last part of the class in a way that differs from the text. Here's a guide to what we're doing in class and how to read along in the text.
In the text, the main definitions of divergence and curl are given as cartesian expressions. In class, we have defined divergence as flux density and from that derived the cartesian expression. In class, we will define curl in terms of a circulation density and from that derive a cartesian expression. In the text, the starting point is the cartesian expression.
With the approach we take in class, the hard work comes in deriving a cartesian expression from the flux density. The Divergence Theorem is almost free (particularly since we are being very casual about some of the limits involved). A similar story happens with curl. The hard work comes in deriving a cartesian expression from the circulation density. Stokes' Theorem is almost free.
Section 27.1 of the text has two theorems that parallel the Divergence Theorem and Stokes' Theorem for vectors fields in the plane. These are called the Planar Divergence Theorem and Green's Theorem. We'll see that Green's Theorem is a special case of Stokes' Theorem and that the Planar Divergence Theorem is an analog of the Divergence Theorem if we treat flux density as the ratio of flux through a curve to the area enclosed by the curve. Note that this is the flux density computed in the Vector Field Analyzer.
Today in class, we defined the curl of a vector field in terms of circulation density. I've written a handout with some details on what we did in class for your reference. I haven't proofread these carefully so it is likely there are typos. Also, this is a revision of an earlier draft from a previous course in which I used some notation different from what we have used in class so there may be places where the notation is inconsistent. In class, I distributed a handout with some exercises on computing curl as a circulation density and as in terms of partial derivatives with respect to cartesian coordinates.
Today in class, we outlined a proof of Stokes' Theorem. I've written a handout with some details on what we did in class for your reference. I haven't proofread these carefully so it is likely there are typos.
Some of you mentioned that you had worked out an expression for the curl in cylindrical coordinates (for use on the project). Here's another handout that outlines an approach to the problem of getting expressions for divergence and curl in cylindrical and spherical coordinates. This is an optional exercise. If you try it and want to check your results, let me know.
Tasha asked a question about a definite integral that came up on Project 3. The answer is a bit subtle but has a good lesson about begin careful with domains and differentiability in using the Fundamental Theorem of Calculus. Here's a handout with some details.
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: