265Lesson19Problems(2)

265Lesson19Problems(2) - t-→ ı 2 sin t-→ 2 t-→ k 0...

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MATH 265: MULTIVARIABLE CALCULUS: LESSON 19 PROBLEMS Problems are worth 20 points each. (1) Sketch the vector field -→ F ( x,y ) = y x 2 + y 2 -→ ı - x x 2 + y 2 -→ by drawing enough vectors at points in the plane to give a sense of the shape of the field. One way to draw a vector field is to select a grid of points, say ( m,n ) with integers m and n between - 3 and 3 (skipping (0 , 0) in this example), and drawing the vector -→ F ( m,n ) with its tail at the point ( m,n ). That sounds like a lot of work, but you should quickly see the pattern that will allow you complete the sketch without a lot of tedious labor. (2) Find a potential function for the vector field -→ F ( x,y,z ) = h z 3 - 2 xy 2 ,z 2 - 2 x 2 y, 3 xz 2 + 2 yz i . (3) Evaluate the scalar line integral Z H ( x 2 + y 2 + z 2 ) d s . where H is the helix given by -→ c ( t ) = 2 cos
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Unformatted text preview: t-→ ı + 2 sin t-→ + 2 t-→ k , 0 ≤ t ≤ 2 π . (4) Evaluate the line integral Z-→ c x 2 d x + xy d y + d z , where-→ c ( t ) = h t,t 2 , 1 i , 0 ≤ t ≤ 1. (5) Evaluate the line integral Z C ( e x-→ ı + xy-→ ) · d-→ s , where C is the curve parametrized by-→ r ( t ) = t-→ ı-t 2-→ , 0 ≤ t ≤ 1. (6) (Bonus question: worth 10 points. Total points for assignment not to exceed 100.) Evaluate the line integral Z C-→ F · d s , where-→ F ( x,y,z ) = e z-→ ı + e x-y-→ + e y-→ k , and C is the path consisting of straight line segments from (0 , , 0) to (0 , , 1) and then from (0 , , 1) to (0 , 1 , 1). 1...
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This note was uploaded on 02/24/2011 for the course MATH 265 taught by Professor Jerrymetzger during the Winter '11 term at North Dakota.

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