FL2009 Final Solutions

# FL2009 Final Solutions - Math 3210 final exam Friday 11...

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Unformatted text preview: Math 3210 final exam Friday 11 December 2009 2:00-4:30 pm Solutions 1. (a) M is the portion of the paraboloid z = x 2 + y 2 sitting over the disc D . (b) Let f : D → R be the function f ( x , y ) = x 2 + y 2 . By a theorem in the notes, ψ ( x , y ) = ( x , y , f ( x , y )) is then an embedding. Therefore M = graph f = ψ ( D ) is a manifold with boundary, of dimension 2, since dim D = 2. (c) Use ψ * μ = p det ( D ψ T D ψ ) dx dy . From D ψ = 1 1 ∂ f ∂ x ∂ f ∂ y we get D ψ T D ψ = 1 + ( ∂ f ∂ x ) 2 ∂ f ∂ x ∂ f ∂ y ∂ f ∂ x ∂ f ∂ y 1 + ( ∂ f ∂ y ) 2 , so det ( D ψ T D ψ ) = 1 + grad f 2 = 1 + 4 ( x 2 + y 2 ) , so ψ * μ = p 1 + 4 ( x 2 + y 2 ) dx dy . (d) area ( M ) = R M μ = R D ψ * μ = R D p 1 + 4 ( x 2 + y 2 ) dx dy . Substitute x = r cos θ , y = r sin θ ; then dx dy = r dr d θ , so area ( M ) = Z 2 π Z 1 p 1 + 4 r 2 r dr d θ = 2 π Z 1 r p 1 + 4 r 2 dr = π 4 Z 4 √ 1 + s ds = π 4 2 3 ( 1 + s...
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## This note was uploaded on 04/14/2010 for the course MATH 3210 at Cornell.

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FL2009 Final Solutions - Math 3210 final exam Friday 11...

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