Question

Linear Algebra

6) Let the surface S be part of the sphere x2 + y² + z² = 4 (oriented away from the origin)that lies within the cylinder x² + y² = 1 and above the plane z = 0, and let denote the unit normal vector in the direction of the orientation. Let C be the boundary curve of S, oriented clockwise when viewed from the x-y plane. Consider the vector field F(x, y, z) = xi + yj + xyzk.

\text { (a) Evaluate } \oint_{C} F \cdot d r \text { without using Stokes' Theorem. }

\text { (b) Evaluate } \iint_{S}(\nabla \times \boldsymbol{F}) \cdot \boldsymbol{n} d S \text { without using Stokes' Theorem. }