${\int }_C\mathbf{F}·d\mathbf{r}$,where C is the intersection of the surface z=$e^{-\left(x^2+y^2\right)}$and the cylinder$x^2+y^2=9$and where

The objective is to find why the statement of Stokes' Theorem requires that the surface S is smooth or piecewise smooth. If this condition is not met, what wrong.

Stokes' Theorem states that,

"Let S be an oriented, smooth or piecewise-smooth surface bounded by a curve C. Suppose that n is an oriented unit normal vector of S and C has a parametrization that traverses C in the counterclockwise direction with respect to n.

if a vector field $\mathbf{F}(x,y,z)=F_1(x,y,z)\mathbf{i}+F_2(x,y,z)\mathbf{j}+F_3(x,y,z)\mathbf{k}$is defined on S, then,

${\int }_C\mathrm{F}(x,y,z)·d\mathrm{r}={\iint }_{S}curl\mathrm{F}(x,y,z)·\mathbf{n}dS$''

Stokes' Theorem is a generalization of Green's Theorem.

Green's Theorem states that,

"Let R be a region in the plane with smooth boundary curve C oriented counterclockwise by $r\left(t\right)=\left(x\right(t),y(t\left)\right)for\le at\le b$if a vector filed

$F(x,y)=(F1(x,y),F2(x,y)isdefinedon$R,then${\int }_C\mathbf{F}·d\mathbf{r}={\iint }_R\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dA=$

If a surface for an intended application of Stokes' Theorem is not smooth or piecewise smooth, there is no guarantee that the incremental application of Green's Theorem in the tangent plane accurate, or even well defined. Hence, the statement of Stokes' Theorem requires that the surface S is smooth or piecewise smooth.