VC5.63

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{\mathrm  {curl}}F={\begin{vmatrix}i&j&k\\{\frac  {\partial }{\partial x}}&{\frac  {\partial }{\partial y}}&{\frac  {\partial }{\partial z}}\\2x^{2}y+yz&2x^{2}y+xz+2yz^{2}&2y^{2}z+xy\end{vmatrix}}\,

=[{\frac  {\partial }{\partial y}}(2y^{2}z+xy)-{\frac  {\partial }{\partial z}}(2x^{2}y+xz+2yz^{2})]i-[{\frac  {\partial }{\partial x}}(2y^{2}z+xy)-{\frac  {\partial }{\partial z}}(2xy^{2}+yz)]j+[{\frac  {\partial }{\partial x}}(2x^{2}y+xz+2yz^{2})-{\frac  {\partial }{\partial y}}(2xy^{2}+yz)]k\,

=[(4yz+x)-(x+4yz)]i-(y-y)j+[(4xy+z)-(4xy+z)]k=0\,

Since {\mathrm  {curl}}F=0\,,it follows that vector field F is conservative.

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