Josephood7

 

Use Stoke's Theorem to evaluate ∫CF⋅dr∫CF⋅dr where F(x,y,z)=xi+yj+8(x2+y2)kF(x,y,z)=xi+yj+8(x2+y2)k and CC is the boundary of the part of the paraboloid where z=25−x2−y2z=25−x2−y2 which lies above the xy-plane and CC is oriented counterclockwise when viewed from above.

Use Stoke's theorem to evaluate ∫∫ScurlF⋅dS∫∫ScurlF⋅dS where F(x,y,z)=−4yzi+4xzj+13(x2+y2)zkF(x,y,z)=−4yzi+4xzj+13(x2+y2)zk and S is the part of the paraboloid z=x2+y2z=x2+y2 that lies inside the cylinder x2+y2=1x2+y2=1, oriented upward.

Use Stokes' Theorem to find the circulation of F⃗ =⟨xy,yz,xz⟩F→=⟨xy,yz,xz⟩ around the boundary of the surface SS given by z=25−x2z=25−x2 for 0≤x≤50≤x≤5 and −4≤y≤4−4≤y≤4, oriented upward. Sketch both SS and its boundary CC.

 The figure below open cylindrical can, SS, standing on the xyxy-plane. (SS has a bottom and sides, but no top.)

 

Three small circles C1C1, C2C2, and C3C3, each with radius 0.20.2 and centered at the origin are in the xy-, yz-, and xz-planes, respectively. The circles are oriented counterclockwise when viewed from the positive z-, x-, and y-axes, respectively. A vector field F⃗ F→ has circulation around C1C1 of 0.09π0.09π, around C2C2 of 0.1π0.1π, and around C3C3 of 5π5π. Estimate curl(F⃗ )curl(F→) at the origin.