If things are working correctly, you should see two pictures below this text. 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 the surface in gray, normal vectors in red, and vector field outputs in blue. The projection of the surface into the xy-plane is outlined in gray. The graphics here aren't perfect but they are better than my drawings.
For each of these, you might find it useful to look at the surface edge-on. From this view, you should be able to estimate the sign of the surface integral. Do you estimate the value of the surface integral to be positive or negative? Compare with the value you calculate.
| Plot for Section 13.5 Problem 23 | Plot for Section 13.5 Problem 29 |
To give credit where it is due, I am 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. We have Mathematica available in several computer labs on campus.