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.
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.
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.
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.
