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Scanned by CamScanner . Find the flux of the vector field F = 2yi - z] + x?k across the surface ofthe parabolic cylinder y = 8x in the first octant bounded by the planesy = 4 and z = 6.[Ans. : 132] |
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Answer» -STEP explanation:Parametrize your SURFACE, say by:r⃗ (u,v):=(u28,u,v),0≤u≤4,0≤v≤6⟹r′u→=(u4,1,0),r′v→=(0,0,1)⟹r′y→×r′v→=∣∣∣∣∣i⃗ u40j⃗ 10k⃗ 01∣∣∣∣∣=(1,−u4,0)Then, since n⃗ =r′y→×r′v→∥∥∥r′y→×r′v→∥∥∥ ( and observe that the NORM of the vectorial product CANCELS in the following!) and also F(r⃗ (u,v))=(2u,−3,u464) , we get that:∬sF⃗ ⋅n⃗ dS⃗ =∬DF⃗ (r⃗ (u,v))⋅(r′y→×r′v→)dA=∫40∫60(2u,−3,u464)⋅(1,−u4,0)dA==∫40∫60114udvdu=664∫40udu=334⋅16=132 |
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