I could use some assistance on a Stokes Theorem problem, I think I have a grasp on what needs to be done but I would appreciate anyone looking it over to make sure I completed the problem properly.
I've been asked to use Stokes' Theorem to find $\int_C F\cdot dr$ where $F(x,y,z)=\langle x^2,4xy^3,xy^2 \rangle$ and C is the rectangular curve that connects, in this order, the points $(0,0,0),(0,3,3),(1,3,3),(1,0,0)$.
I have calculated $curlF=\langle 2xy,-y^2,4y^3 \rangle$
The plane made by the points corresponds to $g(x,y)=y$ so $r(t)=\langle x, 3y, 3y \rangle, \, r_x= \langle 1,0,0 \rangle , \, r_y= \langle 0,3,3 \rangle, \, r_x\times r_y= \langle 0,-3,3 \rangle$
Thus $dS = \langle 0,-3,3 \rangle dydx$
$\begin{align} \int_C F \cdot dr & = \iint curlF \cdot dS \\ & = \int_0^1 \int_0^3 \langle 2xy,-y^2,4y^3 \rangle \cdot \langle 0,-3,3 \rangle dydx \\& =\int_0^1 \int_0^3 3y^2 + 12y^3 dydx \\& =\int_0^1 y^3 + 3y^4 |_0^3 \, dx \\&=\int_0^1270dx \\&=270 \end{align}$
