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MIT18_024S11_soln-pset12

MIT18_024S11_soln-pset12 - Rough Solutions for PSet 12...

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Unformatted text preview: Rough Solutions for PSet 12 1. (12.10:7) We first parameterize the sphere of radius a by r ( , ) = ( a cos sin , a sin sin , a cos ) where 2 and . (We write first so that the normal points outward.) Then the integrals become 2 xy d y d z = a 2 sin 2 cos sin ( a 2 cos sin 2 ) d d , S 2 yz d z d x = a 2 sin sin cos ( a 2 sin sin 2 ) d d , S 2 x 2 d x d y = a 2 cos 2 sin 2 ( a 2 cos sin ) d d . S From here the work is standard. Notice the second and third integral can be combined and simplified. 2. (12.10:12) We want to evaluate S F ndS and here n = ( x, y, z ) since S is a hemisphere. Thus F n = x 2 2 xy y 2 + z 2 . As with the problem above, we parameterize the hemisphere by r ( , ) where / 2 and [0 , 2 ]. So the problem is to evaluate / 2 2 cos 2 sin 2 2 cos sin sin 2 sin 2 sin 2...
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MIT18_024S11_soln-pset12 - Rough Solutions for PSet 12...

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