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Unformatted text preview: D ). (a) In general, how is the integral ZZ S F · d ~ S related to ZZ Φ F · d ~ S . Clearly explain your answer. (b) Let Φ( s,t ) = ( s cos t,s sin t,t ), 0 ≤ s ≤ 1, 0 ≤ t ≤ 2 π . Determine if Φ is positively or negatively oriented. (c) Let F = ( x,y,z2 y ). Evaluate ZZ Φ F · d ~ S . 5. (a) In Chapter 3, we deﬁned the divergence of F at a point ( x,y,z ) as ∇ · F . In class, we demonstrated that this deﬁnition is actually a byproduct of divergence’s original integral deﬁnition. State this deﬁnition and explain how the integral deﬁnition elucidates the physical signiﬁcance of divergence. (b) State the conclusion of Gauss’s Theorem. (c) Calculate the surface integral ZZ S F · d ~ S ; where F = (3 y 2 z 3 , 9 x 2 yz 2 ,4 xy 2 ) where S is the surface of the cube with vertices ( ± 1 , ± 1 , ± 1)....
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This note was uploaded on 02/15/2011 for the course MATH 380 taught by Professor Staff during the Summer '08 term at University of Illinois, Urbana Champaign.
 Summer '08
 Staff
 Math, Calculus

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