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# Use Stokes' Theorem to evaluate $\int_C \mathbf{F} \cdot d \mathbf{r}$, where $\mathbf{F}(x, y, z)=x^2 y \mathbf{i}+\frac{1}{3} x^3 \mathbf{j}+x y \mathbf{k}$ and $C$ is the curve of intersection of the hyperbolic paraboloid $z=y^2-x^2$ and the cylinder $x^2+y^2=1$ oriented counterclockwise as viewed from above.

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We are tasked to use Stoke's Theorem to calculate the integral $\int\limits_C \textbf{F}\cdot d\textbf{r}$.

Stoke's Theorem

Given a vector field $\textbf{F}$ on $\R^3$ with components that have continuous partial derivates and surface $S$ which is an oriented piecewise-smooth surface that is bounded by a simple, close and smooth curve $C$ which is oriented positively, then Stoke's Theorem gives us

$\int\limits_C \textbf{F}\cdot d\textbf{r} = \iint\limits_{S} \text{curl}\textbf{F}\cdot d\textbf{S}$

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