how to use Stokes' Theorem to evaluate following equation: F·dr, F(x,y,z)=(y+sinx)i+(z+cosy)j+xk?

∫c F · dr
= ∫∫s curl F · dS, by Stokes' Theorem
= ∫∫ <-1, -1, -1> · <-z_x, -z_y, 1> dA
= ∫∫ <-1, -1, -1> · <-y, -x, 1> dA, since z = xy
= ∫∫ (x + y - 1) dA.

Since C lies in x^2 + y^2 = 1, convert to polar coordinates:
∫(r = 0 to 1) ∫(θ = 0 to 2π) (r cos θ + r sin θ - 1) * (r dθ dr)
= ∫(r = 0 to 1) (0 + 0 - 2π) r dr
= -π.

I hope this helps!