Answer Happy

F(x,y) = (x2 + y2)*/2 2. Consider the vector field F in the plane given by (q2 +"ya)azi+ (x2 + y2ju23 ) which is defined at every point (6,y) except at the origin (0,0). (a) Find the circulation of F counterclockwise around a circle of constant radius a > 0 centered at the origin by first parameterizing the circle and explicitly computing the line integral f. F. Tds using Change of Variables. (b) Using the result from (b), what's the work done by F counterclockwise around a circle of radius a = 1? What's the work done by F counterclockwise around a circle of radius a = 2? (c) Next, carefully find the curl of F and show that it varies inversely with r'. This will require the quotient rule for differentiation and some algebra. (a) Then, integrate ſa curl F dA over the “annular" (ring-shaped) region R between the unit circle C: 2 + y2 = 1 and the circle Cz: r+ y2 = 22. (See Figure 2.) Use polar coordinates when doing this double integral. (e) As originally stated, Green's Theorem does not apply to this non-simply connected region. However, confirm that in this example culFdA = $ F.Tds - F. Tds. Why does the work? In particular, why did I need to subtract the second integral to make them agree? See the online course notes for a similar example. تجے شری شی Figure 2: The field with annular region R shaded