Which (if any) of the following vector fields are conservative: F rightarrow (x, y) = exy rightarrow + ex+y j rightarrow F rightarrow (x, y) = 2x cos y i rightarrow - x2 sin y j rightarrow Determine if the field F rightarrow (x, y, z) = cos xz, sin yz, xy sin z is conservative. Let F rightarrow(x, y) = xy2 i rightarrow + x2y j rightarrow. Evaluate C F rightarrow · d s rightarrow where C is the upper half of the circle of radius 1 centered at the origin orientated counterclockwise, Find a potential function for F rightarrow (x, y, z) = yz2, xz2 - z, 2xyz - y . Use part (a) to evaluate C Frightarrow · d s rightarrow where C is any path in space from P = (1, 2, 3) to Q = (3, 2, 1). Let F rightarrow (x, y, z) = z2/x i rightarrow + z2/y j rightarrow + 2z In xy k rightarrow. Evaluate intC F rightarrow · d s rightarrow Where C is the path of straight line segments form P = (1, 2, 1) to Q = (4, 1, 7) to R = (5, 11, 7), and then back to P. (Bonus question: worth 10 points. Total points for assignment not to exceed 100.) Suppose T · d s rightarrow = 1 where T is the path from P = (-2, 1) to Q = (2, 1) along the top half of the ellipse 3x2 + 11y2 = 23. Determine the value of · d s rightarrow where B is the path form P to Q along the bottom half of that ellipse.
