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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 $$\vec F = y\,\vec i - x\,\vec j + y{x^3}\,\vec k$$ and $$S$$ is the portion of the sphere of radius 4 with $$z \ge 0$$ and the upwards orientation. Solution
2. Use Stokes’ Theorem to evaluate $$\displaystyle \iint\limits_{S}{{{\mathop{\rm curl}\nolimits} \vec F\centerdot d\vec S}}$$ where $$\vec F = \left( {{z^2} - 1} \right)\,\vec i + \left( {z + x{y^3}} \right)\,\vec j + 6\,\vec k$$ and $$S$$ is the portion of $$x = 6 - 4{y^2} - 4{z^2}$$ in front of $$x = - 2$$ with orientation in the negative $$x$$-axis direction. Solution
3. Use Stokes’ Theorem to evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F = - yz\,\vec i + \left( {4y + 1} \right)\,\vec j + xy\,\vec k$$ and $$C$$ is is the circle of radius 3 at $$y = 4$$ and perpendicular to the $$y$$-axis. $$C$$ has a clockwise rotation if you are looking down the $$y$$-axis from the positive $$y$$-axis to the negative $$y$$-axis. See the figure below for a sketch of the curve.
Solution
4. Use Stokes’ Theorem to evaluate $$\displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}$$ where $$\vec F = \left( {3y{x^2} + {z^3}} \right)\,\vec i + {y^2}\,\vec j + 4y{x^2}\,\vec k$$ and $$C$$ is is triangle with vertices $$\left( {0,0,3} \right)$$, $$\left( {0,2,0} \right)$$ and $$\left( {4,0,0} \right)$$. $$C$$ has a counter 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.
Solution