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midterm.math20e.fall.2005

# midterm.math20e.fall.2005 - Solutions to the Midterm By H...

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Solutions to the Midterm By H˚ akan Nordgren Please note that there are three versions of this midterm (just like there were three versions of the quiz); if the problems that I am solving are not the ones with which you are familiar, then use the same method with the different numbers. Question 1: Give a parameterization of the upper hemisphere of radius 3 and centre (0,0,0), using trig functions. Solution: The upper hemisphere of radius 3 can be parameterized using Φ( u, v ) = (3 cos( u ) sin( v ) , 3 sin( u ) sin( v ) , 3 cos( v )) , when 0 u 2 π and 0 v π . Question 2: Let S be the surface parameterized by Φ( u, v ) = ( u, v, 3 u 2 ) where 0 u 1 and 0 v 1. Let F ( x, y, z ) = ( z, y, x ). Compute the flux of F through S . Solution: To compute the flux of F through S we need to compute Z S F.dS = Z 1 0 Z 1 0 F (Φ( u, v )) . ( Φ ∂u × Φ ∂v ) dudv. We start by figuring out the different pieces of the second integral. The easiest piece is F (Φ( u, v )) = F ( u, y, 3 u 2 ) = (3 u 2 , v, u ). Now we need to compute the normal ( Φ ∂u × Φ ∂v ) . We see that

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midterm.math20e.fall.2005 - Solutions to the Midterm By H...

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