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.
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 in the two problems above. 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 field F=x i+y j. Compare what you see with the results of the first problem above. Try the vector field F=-y i+ x j . Compare what you see with the results of the second problem above.
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.
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.
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.