Chapter 15 - Practice Exercises - Page 910: 40

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Applying Stoke's Theorem, we have $\oint F \cdot dr=\iint _S (\nabla \times F) \cdot n d\sigma$ Here, $(\nabla \times F)=8yi$ and $n=\dfrac{1}{\sqrt 2}j+\dfrac{1}{\sqrt 2}k$ This implies that $(\nabla \times F) \cdot n=0$ Then, we have $\iint _S (\nabla \times F) \cdot n d\sigma=\iint _{R} (0) d\sigma $ This implies that $\iint _S (\nabla \times F) \cdot n d\sigma=0$