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### Section 6-1 : Curl and Divergence

For problems 1 & 2 compute $${\mathop{\rm div}\nolimits} \vec F$$ and $${\mathop{\rm curl}\nolimits} \vec F$$.

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

For problems 3 & 4 determine if the vector field is conservative.

1. $$\displaystyle \vec F = \left( {4{y^2} + \frac{{3{x^2}y}}{{{z^2}}}} \right)\,\vec i + \left( {8xy + \frac{{{x^3}}}{{{z^2}}}} \right)\vec j + \left( {11 - \frac{{2{x^3}y}}{{{z^3}}}} \right)\vec k$$ Solution
2. $$\vec F = 6x\,\vec i + \left( {2y - {y^2}} \right)\vec j + \left( {6z - {x^3}} \right)\vec k$$ Solution