For the following vector field, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume the boundary curve has a counterclockwise orientation. F = {x+y,x² - y), where R= {xy): y² sxs 18- y?} a. The circulation on the boundary is

3. Let D be the upper half-plane (y > 0) and consider the vector field F: D+R² given by F(x, y) = (7, =7). Test whether F is a gradient field, and if so, find a potential P.

(5xy, 5y?) for the region enclosed by the curves y = x? and y = x in the first quadrant, without parameterizing the curves (i.e. use the flux Compute the outward flux of the vector field F version of Green's theorem).

5. Discuss real world quantities that the circulation of a vector field around a closed loop could measure? Now let F (1, 1). Sketch the vector field F at various points on the edges of the square. Based on your sketch, what value would you expect for the circulation of F around the perimeter of the square. Use Stokes' theorem to find the circulation of F for any closed loop C in space. yzi + zxj + yk. Draw a unit square in the xy plane with one corner at the origin and another at