Calculating work done by force field ( multivariable calculus )

Question

Calculate the work done by the force field F(x,y,z)=(x^x +z² )i + (y^y  + x² )j + (z^z +y²)k when a particle moves under its influence around the edge of the part of the sphere x² +y² +z² =4 that lies in the first octant , in a counterclockwise direction as viewed from above.

Answer 1

You can use Stokes' theorem. We wish to calculate

∫∇xF·nds (with n  the unit normal) over the surface

in the 1st octant. ∇xF=2<y,z,x> and n=(1/ρ)<x,y,z>

with ρ=2.

∇xF·n=(2/ρ)(xy+yz+xz), then convert to spherical coordinates and use ds=ρ²Sinφdφdθ. The integrals are a little tedious but I came out with ∫_s=2ρ³=2(2)³=16