Consider the vector field \(\vF = \langle x^2 ,y^2 ,z^2 \rangle\) and the circle \(C_1\) parameterized as \(\vr(t) =\langle \sqrt{2}\cos(t), \sqrt{2}\cos(t), 2\sin(t)\rangle\) for \(0\leq t\leq 2\pi\text{.}\)
Let \(S_1\) be the hemisphere of the sphere of radius \(2\) centered at the origin with \(y\leq x\text{.}\) Calculate the flux of \(\curl(\vF)\) through \(S_1\text{.}\)
Consider the vector field \(\vG = x\vi + y^2z\vj + x^2\vk\) and the curve \(C_2\text{,}\) which is the triangle with vertices \((1,0,0)\text{,}\)\((0,1,0)\text{,}\) and \((0,0,1)\) with orientation corresponding to the order the points are listed here.
The vertices of \(C_2\) lie in a plane. Let \(S_2\) be the portion of this plane lying in the first octant, i.e., the portion with \(x,y,z\geq 0\text{.}\) Find the flux of \(\curl(\vG)\) through \(S_2\text{.}\)
