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Final_chorin

# Final_chorin - Math 53 A.Chorin Mock Final 1 Find the...

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Unformatted text preview: Math 53, A.Chorin, Mock Final 1. Find the maxima and minima of f (m, y, z) = V932 + y2 + 22 with the constraint 4x2 + y2 + % 22 == 1. Sketch the constraint function; interpret your results geomet- rically; specify whether the points you ﬁnd are maxima or minima. Find the local and global maxima and minima of f (:13, y) = (1 + 2.1:)(2: + y) on the square 0 S x S 13, 0 S y S 1. WW. riff, (if ‘ .\E Show that //D‘/1—(cos2<1—%))eydA<%, where D is the triangle with vertices (,0 0), ,). Find the volume inside the sphere 3:2 + y2 + z2 =4 and outside the cylinder Calculate ffD z dV, where E is bounded by the planes :1: = 0, y = 0, z = 0, x+z=1,\$+y=1. Use Green‘s theorem to evaluate fc 2323; dx + mysdy, where C is the boundary of the square with vertices (i1, i1) oriented clockwise. Find dA on the sphere :13 : Rcosqicos 6, y = Rcosqbsin 6, z = Rsin QS. Use it to calculate the surface area of the sphere. Use the result to calculate the volume of the sphere. Use Stokes’ theorem to evaluate ffs curl F - (15, where F = (xyz, :12, 629 cos 2) and S is the hemisphere x2 + y2 + 22 = 1, z 2 O, oriented up. ...
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