I was trying to check if Stokes would hold in a field $(3x^2y^3,-x^3y^2)$ (non conservative as curl is non zero) for the figure below but I got that LHS (the surface integral) was not equal to RHS (line integral) in Stokes law. So does that mean that Stokes law does not always hold?
EDIT: added my attempt
As the curve is clockwise oriented, the normal vector is $(0, 0, -1)$ and the double integral should evaluate to $\displaystyle \small \frac{1352}{3}$.
For the line integral,
$\vec F = (3x^2y^3,-x^3y^2)$ and we have $3$ line segments $y = 1, x \in(3,1); \ y = x, x \in (1, 3); \ x = 3, y \in(3,1)$
$I = \displaystyle \small \int_3^1 (3x^2,-x^3) \cdot (1, 0) \ dx + \int_1^3 (3x^5,-x^5) \cdot (1, 1) \ dx + \int_3^1 (27y^3,-27y^2) \cdot (0, 1) \ dy$
$ = \displaystyle \small \int_3^1 3x^2\ dx + \int_1^3 2x^5 \ dx + \int_3^1 -27y^2 \ dy$
$ = \displaystyle \small -26 + \frac{728}{3} + 234 = \frac{1352}{3}$