Answer Happy

= 2. Consider the cylindrical vector field G(2,0, z) = 1 ê, + lệp +lên. The left panel below shows a view of G slightly offset from the positive z-axis; the rotating nature of the vector field is evident. The right panel shows a different view of the vector field, illustrating the cylindrical paths in (i), (ii), and (iii). Compute the following line integrals using dr = dpệp + pdo êx + dz êz. 2 3 -1 (i) W1 = G. dr where o = 7/4, z = -1, and p changes from 1 to 3 on C1. P Ci (ii) W2 = G. dr where p=3, z = -1, and • changes from 1/4 to 8/2 on C2. C2 be (iii) W3 = G. dr where p= 3,0 = 7/2, and z changes from -1 to 2 on C3. (iv) Optional but recommended Show that G has the equivalent cartesian form 2-Y 2 + y G(x, y, z) = î+ x2 + y2 Construct parametric definitions for the three paths above, then confirm the line integrals using dr i=1,2,3. dt Viti Î + 1 Ê. x2 + y2 1. G(rce). dt, Ci