Verify Stokes' Theorem for the vector field ${\bf F}(x, y, z)= y{\bf i} +x^2{\bf j}+z{\bf k}$ and the part of the paraboloid $z = x^2+y^2$ that lies below the plane $z=1$
Can anyone please verify if I have chosen the right parametrization of the curve: $x^2 +y^2 = 1$ and $z =1$
so, ${\bf r}(t) = \cos(t){\bf i} + \sin(t){\bf j} +{\bf k}$
Then, ${\bf F}({\bf r}(t))= \sin(t){\bf i} + \cos^2(t){\bf j} +{\bf k}$
and $t$ would go from $0$ to $2\pi$
Is my parametrization and limits of $t$ ok? Thank you