Lecture 27 Continued from last class F Gm cososino p o 0 Gm cososino2Rcoso o 0 and now let u coso and u sino o F 2GmR u u 0 GmR 0 These are obviously perpendicular because their dot product equals zero so this is a rectangle Surface integrals and ux Given F F F F Flux F n s How do we nd s We need to parameterize Example 1 Cylinder 0 z Rcos0 Rsin0 z Theta goes from 0 to 2 and z goes from 0 to H What is s for the cylinder 0 Rsin0 0 Rcos0 0 0 z 0 0 z Tangent curves Rsin0 Rcos0 0 and 0 0 1 Area R 0 z s R 0 z Area R 0 z 2 R z 2 RH Example 2 Sphere of Radius R 0 o Rsinocos0 Rsinosin0 Rcoso Theta goes from o to 2 and phi goes from 0 to 0 Rsinosin0 Rsinocos0 0 o Rcosocos0 Rcososin0 Rsino s R sino o 0 Area R sino o 0 R 2 0 4 R Example 3 Graph x y x y u x y x 1 0 u y 0 1 u s u u 1 x y The upward facing unit normal n n s u u 1 x y
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