7. Determine whether the vector field F(x, y) = (eª² co vative. If it is, find its potential function f(x, y). cos y − ² + ½, x³y - e²z², sin y ln x + 2y) is conser-

Use the equation giving the flux of the vector field across the curve to calculate the flux of x + 1 y lã (x + 1)² + y²' (x + 1)² + y² F(x, y) = across C, the segment 7 ≤ y ≤ 9 along the y-axis, oriented upwards. (Use symbolic notation and fractions where needed.) I F. dr =

6. Consider the vector field F(ar, y) = (8r°y+ e")î+ (Ka + e")3, where K is some constant. (a) (5 points) For which value of K is F the gradient of a function? For this value, find f such that F = Vf. (b) (5 points) For the value of K found in part (a), evaluate fe F dr, where C is the segment of the curve y = a from (0,0) to (1, 1). (c) (5 points) If instead K = 1, use Green's Theorem to evaluate f F dr, where C is the boundary of the triangle with vertices (0,0), (2,0), and (2, 1), traversed counterclockwise.

In the images below, the vector field F is shown by the blue arrows and the curve Cis shown in red. Indicate the value of F ·dT. 1. /, F ·dÌ = 0 2. ,F ·dT > 0 3. F dT < 0 4. The value of F ·dT cannot be determined with only this information. X