2. Use Stokes' Theorem to evaluate the line integral $\int_C \vec{F} \cdot d\vec{r}$ and assume $C$ is positively oriented. $\vec{F} = (2z + x, y - z, x + y)$ and $C$ is the triangle with vertices $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$
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The curl of F is given by: curl(F) = (∂Fz/∂y - ∂Fy/∂z, ∂Fx/∂z - ∂Fz/∂x, ∂Fy/∂x - ∂Fx/∂y) In this case, we have: ∂Fz/∂y = 1 ∂Fy/∂z = -1 ∂Fx/∂z = 1 ∂Fz/∂x = 2 ∂Fy/∂x = 1 ∂Fx/∂y = 0 So the curl of F is: curl(F) = (1 - (-1), 1 - 2, 1 - 0) = (2, -1, 1) Show more…
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