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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.
    The curve is a triangle with vertices (0,0,4), (0,2,0) and (4,0,0).  This triangle is completely in the 1st octant and if you are in front of the triangle looking towards the origin It is traced out with a clockwise rotation.
  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.
    This is a sketch with the standard 3D coordinate system.  The positive z-axis is straight up, the positive x-axis moves off to the left and slightly downward and positive y-axis moves off the right and slightly downward.  The circle is centered on the x-axis at x=-3.  If you are standing on the positive x-axis and looking towards the circle there is a counter clockwise orientation to the circle.