VC5.64

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{\mathrm  {curl}}F={\begin{vmatrix}i&j&k\\{\frac  {\partial }{\partial x}}&{\frac  {\partial }{\partial y}}&{\frac  {\partial }{\partial z}}\\2xy-z^{3}&x^{2}+z&y-3xz^{2}\end{vmatrix}}\,

=[{\frac  {\partial }{\partial y}}(y-3xz^{2})-{\frac  {\partial }{\partial z}}(x^{2}+z)]i-[{\frac  {\partial }{\partial x}}(y-3xz^{2})-{\frac  {\partial }{\partial z}}(2xy-z^{3})]j+[{\frac  {\partial }{\partial x}}(x^{2}+z)-{\frac  {\partial }{\partial y}}(2xy-z^{3})]k\,

=(1-1)i+(-3z^{2}+3z^{2})j+(2x-2x)k=0\,

Hence F is conservative.

Let the scalar potential be \phi (x,y,z)\,

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

=F\cdot dr={\frac  {\partial \phi }{\partial x}}dx+{\frac  {\partial \phi }{\partial y}}dy+{\frac  {\partial \phi }{\partial z}}dz=d\phi \,

Therefore,d\phi =F\cdot dr=[(2xy-z^{3})i+(x^{2}+z)j+(y-3xz^{2})k]\cdot (dxi+dyj+dzk)\,

=(2xy-z^{3})dx+(x^{2}+z)dy+(y-3xz^{2})dz\,

=(2xydx+x^{2}dy)-(z^{3}dx+3xz^{2}dz)+(zdy+ydz)\,

=d(x^{2}y)-d(z^{3}x)+d(zy)\,

Therefore,\phi =x^{2}y-z^{3}x+zy+c\,

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