VC5.67

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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&y&z\end{vmatrix}}\,

=[{\frac  {\partial z}{\partial y}}-{\frac  {\partial y}{\partial z}}]i-[{\frac  {\partial z}{\partial x}}-{\frac  {\partial x}{\partial z}}]j+[{\frac  {\partial y}{\partial x}}-{\frac  {\partial x}{\partial y}}]k=0i-0j+ok=0\,

Hence the given vector field is conservative.

Let F=\nabla \phi \, so that \nabla \phi \cdot dr=F\cdot dr\,

=[i{\frac  {\partial \phi }{\partial x}}dx+j{\frac  {\partial \phi }{\partial y}}dy+k{\frac  {\partial \phi }{\partial z}}dz]\cdot (dxi+dyj+dzk)\,

=(xi+yj+zk)\cdot (dxi+dyj+dzk)\,

={\frac  {\partial \phi }{\partial x}}dx+{\frac  {\partial \phi }{\partial y}}dy+{\frac  {\partial \phi }{\partial z}}dz]=xdx+ydy+zdz\,

=d\phi ={\frac  {1}{2}}d(x^{2}+y^{2}+z^{2})\, which implies

\phi ={\frac  {1}{2}}(x^{2}+y^{2}+z^{2})+c\,

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