= Solution
<Stokes theorem> equates the circulation around the induced <boundary orientation> of an <oriented surface> to the normal flux of <curl>:
$$
\boxed{\oint_{\partial S}\mathbf F\cdot d\mathbf x=\int_S(\nabla\times\mathbf F)\cdot d\mathbf S.}
$$
The usual version requires a piecewise smooth oriented surface and a <vector field> continuously differentiable on a neighbourhood of it. We use the upward orientation wherever the surface is a graph; each outer circle is counterclockwise viewed from above, and an inner circle is clockwise.
Write $\rho=(x^2+y^2)^{1/2}$. The <surface of revolution> has $z=\sqrt{\rho^2+1-\lambda}$ with $0\le z\le1$. For $0<\lambda<1$, it is a disk-shaped cap of a <two-sheeted hyperboloid>, running from $\rho=0,z=\sqrt{1-\lambda}$ to $\rho=\sqrt\lambda,z=1$. For $\lambda=1$, it is a <right circular cone>, $z=\rho$, including its apex. For $\lambda>1$, it is an annular strip of a <one-sheet hyperboloid>, between the circles $\rho=\sqrt{\lambda-1},z=0$ and $\rho=\sqrt\lambda,z=1$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ia/paper-3-stokes-surfaces.png]
{title=Disk cap, cone and annular hyperboloid strip in the three positive-parameter regimes of Stokes' theorem}
For $z>0$, differentiation of the graph gives $z_x=x/z$ and $z_y=y/z$, so the upward <vector surface element of a graph> and its scalar area are
$$
\boxed{d\mathbf S=\left(-\frac xz,-\frac yz,1\right)dx\,dy,\qquad
dS=\sqrt{1+\frac{x^2+y^2}{z^2}}\,dx\,dy.}
$$
For $\lambda>1$ this graph expression is singular at the lower circle because the surface has a vertical tangent there, rather than a singular geometric surface. A regular parametrization is
$$
\mathbf X(z,\phi)=(\sqrt{z^2+\lambda-1}\cos\phi,\sqrt{z^2+\lambda-1}\sin\phi,z),\qquad
\mathbf X_z\times\mathbf X_\phi=-\rho\mathbf e_\rho+z\mathbf e_z.
$$
It proves that the graph <surface integral> is interpreted by its integrable endpoint limit. The same parametrization works away from the pole or apex in the other regimes.
Direct differentiation of the first <vector field> gives
$$
\boxed{\nabla\times\mathbf F=(0,0,2).}
$$
Its <surface integral> is twice the projected area, namely $2\pi\lambda$ for $0<\lambda\le1$ and $2\pi[\lambda-(\lambda-1)]=2\pi$ for $\lambda>1$. On an oriented circle of radius $R$, $\mathbf F\cdot d\mathbf x=R^2d\phi$. The outer circulation is $2\pi\lambda$; an inner circle, when present, contributes $-2\pi(\lambda-1)$. Thus
$$
\boxed{\oint_{\partial S}\mathbf F\cdot d\mathbf x=\int_S(\nabla\times\mathbf F)\cdot d\mathbf S=2\pi\min(\lambda,1).}
$$
At $\lambda=1$, excise a small apex circle of radius $\varepsilon$, apply <Stokes theorem> to the smooth truncated cone, and let $\varepsilon\to0$. The removed flux and its inner circulation are both $2\pi\varepsilon^2$, so the limiting equality holds. This avoids silently applying a smooth-surface theorem at the cone's nonsmooth apex.
The second <vector field> is the <azimuthal inverse-radius vector field>, $\mathbf G=\mathbf e_\phi/\rho$. Away from the axis,
$$
\partial_x\left(\frac x{x^2+y^2}\right)=\frac{y^2-x^2}{(x^2+y^2)^2}
=\partial_y\left(\frac{-y}{x^2+y^2}\right),\qquad
\boxed{\nabla\times\mathbf G=\mathbf0\quad(\rho>0).}
$$
The other two <curl> components vanish because the field is independent of $z$ and has zero $z$ component. On any positively oriented circle around the axis, $\mathbf G\cdot d\mathbf x=d\phi$, so its circulation is $2\pi$, independent of radius. Therefore
$$
\boxed{\oint_{\partial S}\mathbf G\cdot d\mathbf x=
\begin{cases}2\pi,&0<\lambda\le1,\\0,&\lambda>1.\end{cases}}
$$
For $\lambda>1$ the surface avoids the axis, so <Stokes theorem> applies directly and returns zero <curl> flux. For $0<\lambda<1$ the disk cap meets the axis, and for $\lambda=1$ the apex lies on it: $\mathbf G$ is undefined there, so the neighbourhood hypothesis of <Stokes theorem> fails. Removing a small circle gives cancelling inner and outer circulations; the inner circulation remains $-2\pi$ as its radius tends to zero. Hence it cannot be discarded as it could for $\mathbf F$. A nonzero circulation here is compatible with zero <curl> away from the excluded axis.
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