# Exam 3 - φ = φ but inside the sphere ρ = ρ assuming that density is constant 6(15 pts Let R be the region in the x y plane bounded by xy = 1 xy

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MATH 253.504–506 An unsupported answer Exam 3, version A is a wrong answer! 12/1/99 This test has 2 pages and 8 problems! Conversion from spherical to cartesian coordinates: x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ dV = ρ 2 sin φdρdθdφ 1. (15 pts.) Evaluate the line integral Z C yds ,whe re C is the line segment starting at (0 , 1 , 1) and ending at (2 , 2 , 3). 2. (10 pts.) Determine the equation of the plane tangent to the surface ~ r ( u, v )= h u 2 + v, u 2 - v, uv i at the point (5 , 3 , - 2). 3. (15 pts.) For the following, determine if they are vector quantities, scalar quantities, or undeﬁned. Here f is a scalar function and ~ F is a vector ﬁeld. (a) (div( ~ F )) (b) curl( f ) (c) curl( ~ F )+div( ~ F ) (d) div( f ) (e) curl(div( ~ F )) 4. (10 pts.) Write a parameterization for the part of the sphere x 2 + y 2 + z 2 = 4 which lies outside of the cylinder x 2 + y 2 = 1. Be sure to speciﬁy the parameter domain. 5. (15 pts.) Find z

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Unformatted text preview: φ = φ but inside the sphere ρ = ρ , assuming that density is constant. 6. (15 pts.) Let R be the region in the x , y plane bounded by xy = 1, xy = 2, xy 2 = 1 and xy 2 = 3. Write Z Z R x dA as an iterated integral in the variables u and v , where u = xy and v = xy 2 . 7. (15 pts.) Use Green’s theorem to evaluate I C ( x 3-y 3 ) dx +( x 3 + y 3 ) dy , where C is the pictured curve, consisting of two circular arcs and two line segments. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Figure for problem 7. 8. (5 pts.) If ~ F is the vector ﬁeld shown, can ~ F be conservative? Why or why not? –1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1 y –1 –0.6 –0.2 0.20.40.60.8 1 x Figure for problem 8....
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## This note was uploaded on 03/26/2008 for the course MATH 251 taught by Professor Skrypka during the Spring '08 term at Texas A&M.

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Exam 3 - φ = φ but inside the sphere ρ = ρ assuming that density is constant 6(15 pts Let R be the region in the x y plane bounded by xy = 1 xy

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