Given that $f(x, y) = (x - 1)^2 y + (y + 1)^2 x$ is a potential function for $\mathbf{F} = \langle 2xy - 2y + (y + 1)^2, (x - 1)^2 + 2yx + 2x \rangle$, calculate integral $\int_{C} \mathbf{F} \cdot d \mathbf{r}$, where $C$ is the lower half of the unit circle oriented counterclockwise.
calculus-iiimultivariable-calculusline-integralsfundamental-theorem-of-line-integralspotential-functionconservative-vector-fieldunit-circlecounterclockwise-orientationevaluate-integralpath-independence
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