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lecture31-09 - 18.02 Multivariable Calculus(Spring 2009...

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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 = S ( F · ˆ n ) dS Shorthand : We often use the notation ˆ n dS = dS , so you could also write flux of F across S = S F · dS. This is a typical surface integral . We already know how to compute one of then, namely the area of a surface: Area ( S ) = 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 S F · dS into a double iterated integral we use (3.2) above and we observe that dS = ˆ n dS = N | N | | N | dA = ( - fx ˆ i - fy ˆ j + ˆ k ) dA 1
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and as a consequence we simply have S F · dS = R F ( x, y, f ( x, y )) · ( - fx ˆ i - fy ˆ j + ˆ k ) dA, where R is the shadow cast by S on the xy -plane. Exercise 1. Consider the vector field F = z ˆ k and the surface S given by the graph of z = x 2 + y 2 above the region R = { ( x, y ) / - 1 x, y 1 } oriented upward. Compute the flux of
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