VC3.15

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

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

=(2z+2x)i-(0-0)j+(-2z-x^{2})k=2(x+z)i-(2z+x^{2})k\,

ii). Similarly as in the above,

\nabla F={\mathrm  {curl}}F={\begin{vmatrix}i&j&k\\{\frac  {\partial }{\partial x}}&{\frac  {\partial }{\partial y}}&{\frac  {\partial }{\partial z}}\\(x^{2}-y^{2})&2xy&(y^{2}-2xy)\end{vmatrix}}\,

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

=(2y-2x-0)i-(-2y-0)j+(2y+2y)k=2(y-x)i+2yj+4yk\,

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