Why is the orientation of S important to the statement of

Stokes’ Theorem? What will change if the orientation is

reversed?

The orientation of $S$is said to be important to the statement of

Stokes’ Theorem.

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

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, localid="1650348783283">∫CF(x,y,z)·dr=∬ScurlF(x,y,z)·ndS".

The orientation of $S$determines the direction of travel along$C$.
In Stokes' Theorem, $C$is traversed in the counterclockwise with respect to $\mathit{n}$, so the orientation of$S$ is important.
If the direction of travel along a curve is reversed, then the sign of integral will change along the curve.
Hence, if the orientation is reversed, then the integral will change sign.