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Section 17.1 : Curl and Divergence

For problems 1 – 3 compute \({\mathop{\rm div}\nolimits} \vec F\) and \({\mathop{\rm curl}\nolimits} \vec F\).

1. \(\vec F = \left( {2y - \cos \left( x \right)} \right)\,\vec i - {z^2}{{\bf{e}}^{3x}}\vec j + \left( {{x^2} - 7z} \right)\vec k\)
2. \(\vec F = - \left( {4y - 1} \right)\,\vec i + x{y^2}\vec j + \left( {x - 3y} \right)\vec k\)
3. \(\displaystyle \vec F = {z^2}\left( {y - x} \right)\,\vec i + \frac{{4{y^2}}}{{{z^3}}}\vec j + \left( {{x^2} - 3z} \right)\vec k\)

For problems 4 – 6 determine if the vector field is conservative.

1. \(\vec F = \left( {2x{y^2} - 16x} \right)\,\vec i + 2y\left( {{x^2} - 1} \right)\vec j + 9\vec k\)
2. \(\vec F = \left( {y - 3z} \right)\,\vec i + \left( {{x^2} + {y^4}} \right)\vec j - 4{z^2}\vec k\)
3. \(\vec F = \left( {18{x^2} + 4{z^3}} \right)\,\vec i - 12yz\,\vec j - \left( {6{y^2} - 12x{z^2}} \right)\vec k\)