265L22 - LESSON 22 Surface Integrals of Vector Fields READ...

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LESSON 22 Surface Integrals of Vector Fields READ: Section 17.5 NOTES: In this section we will be talking about surface integrals RR S g ( x, y, z ) dS of vector fields. We will restrict ourselves to a certain type of surface: ones that have two sides. You might think that all surfaces have two sides, but that is not the case. The M¨ obius strip described on page 980 of the text is a surface with only one side. For a two sided surface, it is possible to paint the surface with say red and blue paint in such a way that we cannot get from the red part to the blue part without crossing a boundary curve of the surface. Such a painting of the M¨ obius strip is not possible. You might want to make a M¨ obius strip for yourself to be really convinced of that surprising fact. As mentioned before, for a line integral, there was a nice geometric interpretation. The value of the line integral gave the area of a curvy fence. There is no such nice geometric interpretation for surface integrals.
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