Hello guys, I'm sorry for taking advantage of you but I need your help. I understand Stroke Theorem but I getting stuck on the following problems:
1. Let C be the intersection curve of the surfaces z =3x-7 and
x^{2} + y^{2} = 1, oriented clockwise as seen from above.
Let F = (4z-1)i + 2xj + (5y + 1)k. Compute the work integral ∫F dr, in two ways:
(a) directly as a line integral
(b) as a double integral, using Stokes' Theorem.
2. Let S be the surface described by z = x^{2} + y^{2} ; z ≤ 4 ; y ≥ 0 and let C be the boundary curve of S with the orientation clockwise.
Let F= < y, z, x >. Find ∫F dr, in two ways:
(a) directly as a line integral
(b) as a double integral, using Stokes' Theorem.
3. Let S be the part of the surface z = 1-x^{2} in the first octant with 0 ≤ y ≤ 2. Let C be the boundary of S, oriented counterclockwise when viewed from above. Let F = <1, 0, y^{2} >. Calculate ∫F dr in two ways:
(a) directly as a line integral
(b) as a double integral, using Stokes' Theorem.