Section 17.5 : 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