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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 17.5 : Stokes' Theorem
- 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.
