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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,z-2 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 by-product 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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