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### Section 6-5 : Stokes' Theorem

1. Use Stokes’ Theorem to evaluate $$\displaystyle \iint\limits_{S}{{{\mathop{\rm curl}\nolimits} \vec F\centerdot d\vec S}}$$ where $$\vec F = {x^3}\,\vec i + \left( {4y - {z^3}{y^3}} \right)\,\vec j + 2x\,\vec k$$ and $$S$$ is the portion of $$z = {x^2} + {y^2} - 3$$ below $$z = 1$$ with orientation in the negative $$z$$-axis direction.
2. Use Stokes’ Theorem to evaluate $$\displaystyle \iint\limits_{S}{{{\mathop{\rm curl}\nolimits} \vec F\centerdot d\vec S}}$$ where $$\vec F = 2y\,\vec i + 3x\,\vec j + \left( {z - x} \right)\,\vec k$$ and $$S$$ is the portion of $$y = 11 - 3{x^2} - 3{z^2}$$ in front of $$y = 5$$ with orientation in the positive $$y$$‑axis direction.
3. Use Stokes’ Theorem to evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F = \left( {z{x^3} - 2z} \right)\,\vec i + xz\,\vec j + yx\,\vec k$$ and $$C$$ is the triangle with vertices $$\left( {0,0,4} \right)$$, $$\left( {0,2,0} \right)$$ and $$\left( {2,0,0} \right)$$. $$C$$ has a clockwise rotation if you are above the triangle and looking down towards the $$xy$$-plane. See the figure below for a sketch of the curve. 4. Use Stokes’ Theorem to evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F = {x^2}\,\vec i - 4z\,\vec j + xy\,\vec k$$ and $$C$$ is is the circle of radius 1 at $$x = - 3$$ and perpendicular to the $$x$$-axis. $$C$$ has a counter clockwise rotation if you are looking down the $$x$$-axis from the positive $$x$$-axis to the negative $$x$$-axis. See the figure below for a sketch of the curve. 