Say $F = [z^{3}, x^{3}, y^{3}]$ and circle is at $x = 2, y^{2} + z^{2} = 9$
So I'm trying to find: $$\int \int curlF \cdot n dA$$
So I understand that $C$ is the boundary curve of a disk and the circle is in the $y-z$ plane. A normal vector is in the $x$ direction so the normal vector is $n = [1,0,0]$
so $$curl F = [3x^{2}, 3y^{2}, 3z^{2}]$$
$$curl F \cdot n = [3x^{2}, 3y^{2}, 3z^{2}] \cdot [1,0,0] = 3y^{2}$$
To convert to polar coordinates:
$$x = z$$ $$y = u \cos{v}$$ $$z = y \sin{v}$$
And then the integral becomes:
$$\int_0^{3} \int_0^{2\pi} 3u^{2}(cos^{2}v)ududv$$
Why is the polar coordinate transformation necessary? Why does $x = z$? Why does $y = \cos(\theta)$? I don't understand the limits of integration of the new integral... can someone explain that?