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21D-FQ02 - 0(0 1 0(0 2 and let C be the closed curve that...

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MAT 021D Fall 02 Final 1. (10 pts.) Let C 1 be a closed curve in 3 space bounding a surface S 1 and let S 2 be a closed surface in 3 space bounding a volume V 2 . Let F be a vector field defined everywhere in three space. a. State the conclusion of the Divergence Theorem. b. State the conclusion of Stokes Theorem. 2. (20 pts.) Let f ( x, y )= e x cos y and F ( x, y )= x 2 y i +( x 3 - y 3 ) j . a. Compute f . b. Compute 2 f . c. Compute ∇· F . d. Compute ∇× F . 3. (20 pts.) Let A be the upper half disk of radius 1 in the x, y plane and let C be the curve bounding A . The curve C has two parts, C 1 the line segment on the x -axis from x = - 1to x = 1 and C 2 the upper half circle of radius 1. Let F =2 x i +2 y j and let n be the outward pointing normal to A . a. Compute Z C 1 F · n ds b. Compute Z C 2 F · n ds c. Compute Z A ∇· F dA 4. (20 pts.) Let S be the triangle in three space with vertices at (1
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Unformatted text preview: , , 0) , (0 , 1 , 0) , (0 , , 2) and let C be the closed curve that bounds it. This triangle lies in the plane 2 x + 2 y + z = 2. a. Let f ( x, y, z ) = z , compute Z S f ( x, y, z ) dS. b. Let F = z 2 k , compute Z C F · d r 1 5. (20 pts.) Let F =-y i + x j x 2 + y 2 a. Compute ∇ × F . b. Let C 1 be a closed curve that does not go around the z axis. Compute I C 1 F · T ds c. Let C 2 be a closed curve that goes around the z axis once in the counter-clockwise direction. Compute I C 2 F · T ds 6. (10 pts.) Suppose F , G are conservative vector fields defined in all of 2 space. Show that F + G is conservative. 2...
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21D-FQ02 - 0(0 1 0(0 2 and let C be the closed curve that...

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