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Mathematics 317 Midterm#3 _2009_1

Mathematics 317 Midterm#3 _2009_1 - surface with boundary...

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Mathematics 317 Midterm #3: Friday, November 20, 2009 1. Answer Question #1. Do only one of Questions #2 and #3. Each question is of equal value. No notes, books or calculators are permitted. 2. Time Limit: 50 minutes 1. Consider the vector field given by r F ) ( ρ f = where , , r k j i r = + + = ρ z y x and ) ( ρ f is a continuous function of . 0 , ρ ρ Let σ be the triangular surface with vertices (1,1,0), (1,0,1) and (1,1,1). (a) Set up a double integral ( do not evaluate ) to find the magnitude of the flux of F through the triangular surface σ . (b) Evaluate the magnitude of the flux of F through the triangular surface σ in the case when . 1 ) ( = ρ f ANSWER ONLY ONE OF THE NEXT TWO QUESTIONS. FOR EACH OF THESE QUESTIONS THE DIVERGENCE THEOREM MAY BE OF USE. 2. Consider the vector field given by k j i F ) 1 ( ) ( ) ( 2 3 2 + + + + + - = z y z y xy z x β α where α and β are constants. Let C be the closed curve bounded by the straight line joining (-1,0,0) to (1,0,0) and the semi-circle . 0 , 0 , 1 2 2 = = + y z y x Let σ be a smooth open
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Unformatted text preview: surface with boundary C . (a) Find the values of the constants α and β so that the flux of F through σ has the same value for any choice of σ . (b) For these special values of the constants α and β , calculate the flux . ∫∫ ⋅ σ S F d [Hint: Pick a convenient σ .] 3. The base of a solid (of constant density) is the disk 25 2 2 ≤ + y x in the plane z = 4. The solid has volume V = 60 and its centroid is located at (2, 1, 10). Consider the vector field given by . ) ( ) ( ) ( 2 2 3 3 1 2 k j i F z y x yz e x x yz + +-+ + + = Find the flux of F (a) outward through the surface of the solid; (b) through the surface of the solid excluding the base. [Hint: the centroid of a solid occupying region R of volume V is given by .] , , 1 ) , , ( = ∫∫∫ ∫∫∫ ∫∫∫ R R R zdV ydV xdV V z y x...
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