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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 17.6 : Divergence Theorem
- Use the Divergence Theorem to evaluate \( \displaystyle \iint\limits_{S}{{\vec F\centerdot d\vec S}}\) where \(\vec F = \left( {3x - z{x^2}} \right)\,\vec i + \left( {{x^3} - 1} \right)\,\vec j + \left( {4{y^2} + {x^2}{z^2}} \right)\,\vec k\) and \(S\) is the surface of the box with \(0 \le x \le 1\), \( - 3 \le y \le 0\) and \( - 2 \le z \le 1\). Note that all six sides of the box are included in \(S\).
- Use the Divergence Theorem to evaluate \( \displaystyle \iint\limits_{S}{{\vec F\centerdot d\vec S}}\) where \(\vec F = 4x\,\vec i + \left( {1 - 6y} \right)\,\vec j + {z^3}\,\vec k\) and \(S\) is the surface of the sphere of radius 2 with \(z \ge 0\), \(y \le 0\) and \(x \ge 0\). Note that all four surfaces of this solid are included in \(S\).
- Use the Divergence Theorem to evaluate \( \displaystyle \iint\limits_{S}{{\vec F\centerdot d\vec S}}\) where \(\vec F = - xy\,\vec i + \left( {z - 1} \right)\,\vec j + {z^3}\,\vec k\) and \(S\) is the surface of the solid bounded by \(y = 4{x^2} + 4{z^2} - 1\) and the plane \(y = 7\). Note that both of the surfaces of this solid included in \(S\).
- Use the Divergence Theorem to evaluate \( \displaystyle \iint\limits_{S}{{\vec F\centerdot d\vec S}}\) where \(\vec F = \left( {4x - {z^2}} \right)\,\vec i + \left( {x + 3z} \right)\,\vec j + \left( {6 - z} \right)\,\vec k\) and \(S\) is the surface of the solid bounded by the cylinder \({x^2} + {y^2} = 36\) and the planes \(z = - 2\) and \(z = 3\) . Note that both of the surfaces of this solid included in \(S\).