2011_Tutorial_9 - 2 3 Ans 4 √ 14 3 4 Compute the work...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 9 1. Find the area of the surface consisting of the part of the sphere of radius 2 centered at origin that lies above the horizontal plane z = 1. (Equation of this sphere is given by x 2 + y 2 + z 2 = 2 2 .) Ans : 4 π 2. Let F ( x,y,z ) = 2 xy i + ( x 2 + 2 yz ) j + y 2 k . Show that F is a conservative vector field. Find a function f such that f = F . Ans : f ( x,y,z ) = x 2 y + y 2 z + K 3. Evaluate Z C g ( x,y,z ) ds , where g ( x,y,z ) = x 2 - yz + z 2 and C is the line segment from (0 , 0 , 0) to (1
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Unformatted text preview: , 2 , 3). Ans : 4 √ 14 / 3 4. Compute the work done by the force F ( x,y,z ) = yz i + 2 y j-x 2 k on a particle that moves along the curve C given by the vector function r ( t ) = t i + t 2 j + t 3 k , for 0 ≤ t ≤ 1. Ans : 17 / 30 5. Evaluate Z C 2 xy dx + ( x 2 + z ) dy + y dz , where C consists of two line segments: C 1 from (0 , , 0) to (1 , , 2), and C 2 from (1 , , 2) to (3 , 4 , 1). Ans : 40...
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