Let the vector v = vi + vj + vak be a differentiable vector function of Cartesian coordinates x, y and z. The curl of the vector v is given by curl v =

For vectors F = 3xy î − y² ĵ and R = x î + y ĵ, the value of ∮C F.dR on the curve C (y = 2x²) in the x-y plane from (0, 0) to (1, 2) is

Directional derivative of f(x, y, z) = xy² + yz³ at the point (2, -1, 1) in the direction of vector î + 2ĵ + 2k (rounded off to two decimal places) is.