91Ó°ÊÓ

According to Stokes' Theorem,

"Assume S is an oriented, smooth or piecewise-smooth surface bound by a curve C.". Assume that n is an oriented unit normal vector of S and that C has a parametrization that traverses C counterclockwise with respect to n.

If F is a vector field defined on S, then,

${∫}_{\mathrm{C}}\mathbf{F}(x,y,z)×d\mathbf{r}={âˆ\neg }_{S}curl\mathbf{F}(x,y,z)×\mathbf{n}d{S}^{-}$

As a result, the hypothesis of the stokes theorem is as follows:

(1) On an open set containing the surface S bounded by C, the vectors field F is defined and continuously differentiable.

(2) The boundary curve C is straightforward, smooth, and closed.

(3) The surface S is smooth and oriented.

The surface S is the portion of the cone $z=\sqrt{x^2+y^2}$inside the cylinder $x^2+y^2=k^2$for some $kâ‰\yen 0$

The goal is to demonstrate that Stokes' Theorem does not apply in this case.

Here, notice that, a cone with equation $z=\sqrt{x^2+y^2}$inside the cylinder $x^2+y^2=k^2$for some $kâ‰\yen 0$is not smooth or piece-wise smooth surface because of vertex $(0,0,0)$.

The normal vector is 0 at the vertex.,

As a result, this surface is not smooth or piecewise smooth.

This implies that Stokes' Theorem does not apply in this case The surface S is smooth and oriented