Verified answer

curl F = (1 - 0, -(0 - 1), 2) = (1, 1, 2).

As for the surface, the sphere and cone intersect when x^2 + y^2 = 1/2

==> z = √2/2. In spherical coordinates (with ρ = 1), this corresponds to φ = π/4.

Thus, we can parameterize the surface as

r(u, v) = (cos u sin v, sin u sin v, cos v) for u in [0, 2π] and v in [0, π/4].

r_u x r_v = (-cos u sin^2(v), -sin u sin^2(v), -sin v cos v)

To get the orientation correct (so the normal points outward), negate this.

n = (cos u sin^2(v), sin u sin^2(v), sin v cos v)

Therefore Stokes' Theorem yields

∫c F · dr

= ∫∫ curl F · dS

= ∫∫ curl F · n dA

= ∫∫ (1, 1, 2) · (cos u sin^2(v), sin u sin^2(v), sin v cos v) dA

= ∫(v in [0, π/4]) ∫(u in [0, 2π]) (cos u sin^2(v)+ sin u sin^2(v) + 2 sin v cos v) du dv

= ∫(v in [0, π/4]) (0 + (2π) 2 sin v cos v) dv

= 2π sin^2(v) {for v = 0 to π/4}

= π.

I hope this helps!

F=e^-xi+e^xj+e^zk=> F*dr=(e^-x)dx+(e^x)dy+(e^z)dz a million) in the x-y plane: z=0=>c is y=6-6x Sc F*dr= a million S[(e^-x)-6e^x]dx=7-6e-a million/e 0 2) in the y-z plane: x=0=>z=a million-y/6 ScF*dr= 0 S[a million-(e^[a million-y/6])/6]dy=-7+e 6 3) in the x-z plane: y=0=>z=a million-x ScF*dr= a million S[(e^-x)-e^(a million-x)]dx=[(a million-e)^2]/e 0 => ScF*dr= [(a million-e)^2]/e+e-7+7-6e-a million/e= -2-4e= -12.87

F=e^-xi+e^xj+e^zk=> F*dr=(e^-x)dx+(e^x)dy+(e^z)dz a million) interior the x-y plane: z=0=>c is y=6-6x Sc F*dr= a million S[(e^-x)-6e^x]dx=7-6e-a million/e 0 2) interior the y-z plane: x=0=>z=a million-y/6 ScF*dr= 0 S[a million-(e^[a million-y/6])/6]dy=-7+e 6 3) interior the x-z plane: y=0=>z=a million-x ScF*dr= a million S[(e^-x)-e^(a million-x)]dx=[(a million-e)^2]/e 0 => ScF*dr= [(a million-e)^2]/e+e-7+7-6e-a million/e= -2-4e= -12.87