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Unformatted text preview: 18.02 Multivariable Calculus (Spring 2009): Lecture 31 Surface Area and flux across cylinders and spheres. April 24 Reading Material: From Simmons : 21.4. From Lecture Notes : V8, V9. Last time: Surface Area. Vector fields in 3D. Surface Integrals and flux. Today: Surface Area and flux across cylinders and spheres. In the last class we learned that the flux across an oriented surface S is flux of ~ F across S = ZZ S ( ~ F n ) dS Shorthand : We often use the notation ndS = d ~ S , so you could also write flux of ~ F across S = ZZ S ~ F d ~ S. This is a typical surface integral . We already know how to compute one of then, namely the area of a surface: Area ( S ) = ZZ S 1 dS so it will not be difficult to generalize this formula. 2 Evaluating flux Integrals Assume S is a surface given by the graph of a function f ( x,y ). Assume S is oriented by using the upward normal. To convert the flux integral ZZ S ~ F d ~ S into a double iterated integral we use (3.2) above and we observe that d ~ S = ndS = ~ N | ~ N | | ~ N | dA = (- fx i- fy j + k ) dA 1 and as a consequence we simply have ZZ S ~ F d ~ S = ZZ R ~ F ( x,y,f ( x,y )) (- fx i- fy j + k ) dA, where R is the shadow cast by S on the xy-plane....
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