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Use Stokes' Theorem to evaluate CFdr\int_C \mathbf{F} \cdot d \mathbf{r}, where F(x,y,z)=x2yi+13x3j+xyk\mathbf{F}(x, y, z)=x^2 y \mathbf{i}+\frac{1}{3} x^3 \mathbf{j}+x y \mathbf{k} and CC is the curve of intersection of the hyperbolic paraboloid z=y2x2z=y^2-x^2 and the cylinder x2+y2=1x^2+y^2=1 oriented counterclockwise as viewed from above.


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Answered 1 year ago
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We are tasked to use Stoke's Theorem to calculate the integral CFdr\int\limits_C \textbf{F}\cdot d\textbf{r}.

Stoke's Theorem

Given a vector field F\textbf{F} on R3\R^3 with components that have continuous partial derivates and surface SS which is an oriented piecewise-smooth surface that is bounded by a simple, close and smooth curve CC which is oriented positively, then Stoke's Theorem gives us

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

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