265Lesson19Problems(2)

265Lesson19Problems(2) - t- + 2 sin t- + 2 t- k , 0 t 2 ....

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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- + 2 sin t- + 2 t- k , 0 t 2 . (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- + xy- ) d- s , where C is the curve parametrized by- r ( t ) = t- -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- + 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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