(a) Evaluate f. (sin x + 3y²)dx + (2x – e-y*)dy, where C is the boundary of the half-disk x² + y? < a?,y 2 0, oriented counterclockwise. (b) Evaluate f. (x² – xy)dx + (xy – y²)dy clockwise around the triangle with vertices (0, 0), (1, 1) and (2, 0).

If the temperature at the point (x, y, z) is given by T (x, y, z) = 70 + Se (x² +y), find the direction from the point (-1, 1, 5) in which the temperature decreases most rapidly. -2 OA. (-540, - 78e*,-74e*) OB. (-16e*, 8e*, 4e*) Oc. (54e*, 78e*, 74e*) OD. (16e*, -8e*, - 4e*)

Let F = (3y + 2) i + (4x – 22) 3 + (3r – y) k Find So FdF where C is a line segment 8. from A(2, –1,2) to B = (1, –1, 3).

Calculate ſc (4(x² − y)i + 8(y² + x)j) · dr it: (a) C is the circle (x − 8)² + (y – 5)² = 1 oriented counterclockwise. Sc (4(x² − y)i +8(y² + x)}) · dr = (b) C is the circle (x − a)² + (y − b)² = R² in the xy- plane oriented counterclockwise. Sc (4(x² − y)i +8(y² + x)}) · dř = ¯