317 hw3.pdf - Math 317 2019W2 Homework 3 Name ~ 1 Let F(x y = hx \u2212yi(a Sketch the vector field(draw at least 15 vectors(b Suppose that at time t = 0 a

317 hw3.pdf - Math 317 2019W2 Homework 3 Name ~ 1 Let F(x y...

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Math 317, 2019W2 Homework 3 Name 1. Let~F(x, y) =hx,-yi.(a) Sketch the vector field (draw at least 15 vectors).(b) Suppose that at timet= 0, a particle is placed at the point (1,2) on some fluidwith velocity field~F(x, y). Find the location of the particle~r(t) at timet.(c) The particle moves along a certain curve. Find an equation for this curve in termsofxandy.(d) (Optional, do not hand in) Do the above problems for~F(x, y) =hy, xi.2. LetCbe the line segment from (0,3) to (2,3). ComputeZCxeydx+x2ydy.3. Evaluate the line integralsIj=ZCj(x+ 2y)dx+x2dy,j= 1,2, whereC1is the linesegment from (0,0) to (4,0), andC2consists of line segments from (0,0) to (2,1) andfrom (2,1) to (4,0). IsI1equal toI2?4. Evaluate the line integralI=ZCz2dx+x2dy+y2dz, whereCis the line segment from(1,0,0) to (-2,1,2).5. Evaluate the line integralI=ZC~F·d~r, where~F=xbı+yb+yzbkandCis given bythe vector function~r(t) = costbı+ sintb+tbk, 0tπ.6. Determine if~F=hx3,-y4iis conservative or not. If conservative, find a potential.7. Determine if

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