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Let $\vec{F}=\langle x^2,y^2,-z\rangle$. I want calculate the line integral of $F$ on curve $C$ which is a closed loop consisting of the line segments that form the triangle with vertices $(0,0,0), (0,2,0),(0,0,2)$.

Since $C$ is a simple, closed, smooth curve with positive orientation that encloses an oriented smooth surface, the Stokes' Theorem gives $$\oint_C\vec{F}\cdot d\vec{r}=\iint_S\text{curl}\vec{F}\cdot d\vec{S}$$ but $$\text{curl}\vec{F}=\vec{0}$$ since it's a conservative vector field (there exists its potential function). Thus $$\oint_C\vec{F}\cdot d\vec{r}=\iint_S\vec{0}\cdot d\vec{S}=0.$$ Is it correct?

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Yes, everything is fine, well-done.

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