1. Is F = (yz, xz + z, xy+2yz) a gradient field? 2. If (y, x,0) a gradient field? 3. Find a vector field such that curl F = (2,-3,4). 4. Does = (2x, 3y-22,-52) have a vector potential? If so, find one. 5. Does G = (x2,y2, 22) have a vector potential? If so, find one. 6. Use Stokes' Theorem to prove Green's Theorem. More precisely, show that if u(x, y) and v(x, y) are functions of x and y and C' is a closed curve in the ry-plane oriented counterclockwise, then dr: dr dy, where R is the region in the xy-plane enclosed by C. 7. An electric charge q at the origin produces and electric field E =973 (a) Is curl E? (b) Does E satisfy the curl test? (c) Is E a gradient field? 8. A thin wire along the z-axis carrying a current I produces a magnetic field 21 B-2 (+220) where c is the speed of light. (a) Is curl B=0? C (b) Does B satisfy the curl test? (c) Is B a gradient field?