Question

1 point) Use Stoke's Theorem to evaluate ∫CF⋅dr∫CF⋅dr where F(x,y,z)=xi+yj+1(x2+y2)kF(x,y,z)=xi+yj+1(x2+y2)k and CC is the boundary of the part of the paraboloid where z=81−x2−y2z=81−x2−y2 which lies above the xy-plane and CC is oriented counterclockwise when viewed from above.

Answer 1

0

Explanation:

Thinking process:

`\int\limits^a_b {cF} \, .dr`

=
`\int\limits^a_b {x} \, dx \int\limits^a_b {x} \, dx curlFdS` by Stoke's Theorem

=
`\int\limits^a_b {} \, \int\limits^a_b {} \, < 12y, -12x, 0 > .< z_x,-z_y, 1 > dA\\= \int\limits^a_b {} \, \int\limits^a_b {} \, < 12y, -12x, 0 > .< 2x, 2y, 1 > dA` since z =
`25-x^(2) -y^(2) \\= 0`