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11.30 - Section 13.7 wrap-up Go through Stewart's...

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Section 13.7 wrap-up: Go through Stewart’s derivation of the formula ∫∫ S F d S = ∫∫ S F n dS = ∫∫ D F ( r u × r v ) dA on page 781. Discuss Group Work 2: 1. ⟨∂ G / x , G / y , G / z = 2 x ,2 y ,2 z 2. Outward 3. n = (1/ R ) x , y , z (check: | n | 2 = (1/ R 2 ) ( x 2 + y 2 + z 2 ) = 1) 4. Everywhere but (0,0,0) 5. On the surface x 2 + y 2 + z 2 = R 2 , we have F = ( x i + y j + z k )/ R 3 = (1/ R 3 ) x , y , z so F n = (1/ R 4 ) ( x 2 + y 2 + z 2 ) = 1/ R 2 and ∫∫ S F n dS = ∫∫ S 1/ R 2 dS = (1/ R 2 ) Area( S ) = (1/ R 2 ) (4 π R 2 ) = 4 π . Check this with spherical coordinates: Imitating Example 4, we use r ( φ , θ ) = R sin φ cos θ i + R sin φ sin θ j + R cos φ k with 0 φ π and 0 θ 2 π ; note that the extra factor
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of R will multiply r φ and r θ by R, so that r φ × r θ = R 2 (sin 2 φ cos θ i + sin 2 φ sin θ j + sin φ cos φ k ) while F ( r ( φ , θ )) = (1/( R 2 ) 3/2 ) (( R sin φ cos θ ) i + ( R sin φ sin θ ) j + ( R cos φ ) k ) = (1/ R 2 ) ((sin φ cos θ ) i + (sin φ sin θ ) j + (cos φ ) k ) so F ( r φ × r θ ) = sin 3 φ cos 2 θ + sin 3 φ sin 2 θ + sin φ cos 2 φ
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