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In Exercises $5-14,$ use Green's Theorem to find the counterclockwise circulation and outward flux for the field $F$ and curve $C .$ $$ \begin{array}{l}{\mat...

Question

In Exercises $5-14,$ use Green's Theorem to find the counterclockwise circulation and outward flux for the field $F$ and curve $C .$ $$ \begin{array}{l}{\mathbf{F}=\left(y^{2}-x^{2}\right) \mathbf{i}+\left(x^{2}+y^{2}\right) \mathbf{j}} \\ {C : \text { The triangle bounded by } y=0, x=3, \text { and } y=x}\end{array} $$

In Exercises $5-14,$ use Green's Theorem to find the counterclockwise circulation and outward flux for the field $F$ and curve $C .$ $$ \begin{array}{l}{\mathbf{F}=\left(y^{2}-x^{2}\right) \mathbf{i}+\left(x^{2}+y^{2}\right) \mathbf{j}} \\ {C : \text { The triangle bounded by } y=0, x=3, \text { and } y=x}\end{array} $$



Answers

In Exercises $7-12,$ use the surface integral in Stokes' Theorem to calculate the circulation of the field $\mathbf{F}$ around the curve $C$ in the indicated direction.
\begin{equation}
\begin{array}{l}{\mathbf{F}=x^{2} \mathbf{i}+2 x \mathbf{j}+z^{2} \mathbf{k}} \\ {C : \text { The ellipse } 4 x^{2}+y^{2}=4 \text { in the } x y \text { -plane, counterclockwise }} \\ {\text { when viewed from above }}\end{array}
\end{equation}


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