VC5.75

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

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

=(-x+x)i-(-y+y)j+(-z+z)k=0\,

Hence the given vector field is irrotationa. Let F=\nabla \phi \,

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

Therefore {\frac  {\partial \phi }{\partial x}}=x^{2}-yz\, when \phi ={\frac  {x^{3}}{3}}-xyz+f_{1}(y,z)\, --(1)

{\frac  {\partial \phi }{\partial y}}=y^{2}-zx\, when \phi ={\frac  {y^{3}}{3}}-xyz+f_{2}(z,x)\, --(2)

{\frac  {\partial \phi }{\partial z}}=z^{2}-xy\, when \phi ={\frac  {z^{3}}{3}}-xyz+f_{3}(x,y)\, --(3)

(1),(2),(3)each represent phi.These agree if we choose f_{1}(y,z)={\frac  {y^{3}}{3}}+{\frac  {z^{3}}{3}},f_{2}(z,x)={\frac  {x^{3}}{3}}+{\frac  {z^{3}}{3}},f_{3}(x,y)={\frac  {x^{3}}{3}}+{\frac  {y^{3}}{3}}\,

Therefore \phi ={\frac  {x^{3}+y^{3}+z^{3}}{3}}-xyz+c\,,c being constant.

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