Determine whether the vector fields that follow are conservative. If the field is conservative, find a potential function for it

$\mathbf{G}(x,y,z)=y^{2}z\mathbf{i}+2xyz\mathbf{j}+x{y}^2\mathbf{k}$

The given expression is$\mathbf{G}(x,y,z)=y^{2}z\mathbf{i}+2xyz\mathbf{j}+x{y}^2\mathbf{k}$

Consider the vector field

$\mathbf{G}(x,y,z)=y^{2}z\mathbf{i}+2xyz\mathbf{j}+x{y}^2\mathbf{k}$

If $curl\overrightarrow{G}=0$,then field is conservative

now,

$curl\overrightarrow{G}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ \left(y^{2}z\right)& \left(2xyz\right)& x{y}^2\end{array}\right|$

$=\mathbf{i}\left(\frac{\partial \left(x{y}^2\right)}{\partial y}-\frac{\partial \left(2xyz\right)}{\partial z}\right)-\mathbf{j}\left(\frac{\partial \left(x{y}^2\right)}{\partial x}-\frac{\partial \left(y^{2}z\right)}{\partial z}\right)+\mathbf{k}\left(\frac{\partial \left(2xyz\right)}{\partial x}-\frac{\partial \left(y^{2}z\right)}{\partial y}\right)$

$=\mathbf{i}(2xy-2xy)-\mathbf{j}\left(y^2-y^2\right)+\mathbf{k}(2yz-2yz)$

$=\mathbf{i}\left(0\right)-\mathbf{j}\left(0\right)+\mathbf{k}\left(0\right)$

$=0$

Therefore it is conservative

To find potential function

$\mathbf{G}(x,y,z)=\nabla g(x,y,z)$

$=\frac{\partial g}{\partial x}\mathbf{i}+\frac{\partial g}{\partial y}\mathbf{j}+\frac{\partial g}{\partial z}\mathbf{k}$

here,

$\frac{\partial g}{\partial x}=g_x=y^{2}z$

$\frac{\partial g}{\partial y}=g_y=2xyz$

$\frac{\partial g}{\partial z}=g_z=x{y}^2$

$\text{Integrate}{g}_x\text{with respect to}x\text{,}$

$g(x,y,z)=x{y}^{2}z+B\left(y\right)\dots \dots \left(1\right)$

$\text{Here,}B\text{is a constant with respect to}x$

$\text{Then differentiate}g(x,y,z)\text{with respect to}y\text{,}$

$g_y(x,y,z)=2xyz+B^{\text{'}}\left(y\right)$

$\text{Compare this with}{g}_y=2xyz$

$B^{\text{'}}\left(y\right)=0$

$\text{Integrate this with respect to}y\text{,}$

$\text{Thus,}B\left(y\right)=0+h\left(z\right)$

$\text{Now substitutingthis value in (1)}$

$g(x,y,z)=x{y}^{2}z+h\left(z\right)\dots \dots \left(2\right)$

$\text{Differentiating(2) with respect to}z\text{,}$

$g_z(x,y,z)=x{y}^2+h^{\text{'}}\left(z\right)$

$\text{Comparingthis with}{g}_z=x{y}^2$

$h^{\text{'}}\left(z\right)=0$

$\text{nowintegratingthis with respect to}z\text{.}$

$h\left(z\right)=\int h^{\text{'}}\left(z\right)dz=0$

$\text{Substitutingthis value in (2).}$

$g(x,y,z)=x{y}^{2}z$

$\text{Therefore, the required potential function is}$$g(x,y,z)=x{y}^{2}z$