Give an example of a non-conservative vector field whose divergence is uniformly equal to zero in ${\mathrm{ℝ}}^3\text{.}$.

The objective is to give an example of a non-conservative vector field whose divergence is uniformly equal to zero in ${\mathrm{ℝ}}^3$

A vector field $F(x,y,z)=\left\{F_1\right(x,y,z),F_2(x,y,z),F_3(x,y,z\left)\right\}$is conservative, if and only if the following conditions are satisfied:

$\frac{\partial F_1}{\partial y}=\frac{\partial F_2}{\partial x},\frac{\partial F_1}{\partial z}=\frac{\partial F_3}{\partial x},\frac{\partial F_3}{\partial y}=\frac{\partial F_2}{\partial z}$

Consider the following vector field,

$F(x,y,z)=(y+z,2x+xz+2xy+y)$

For this vector field,

$F_1(x,y,z)=y+z,F_2(x,y,z)=2x+xz,F_3(x,y,z)=2xy+y$

Notice that,

$\frac{\partial F_3}{\partial y}=\frac{\partial }{\partial y}(2xy+y)$

$=2x+1$

$\frac{\partial F_2}{\partial z}=\frac{\partial }{\partial z}(2x+xz)$

$=x.$

Hence in this case$\frac{\partial F_3}{\partial y}\not\equiv \frac{\partial F_2}{\partial z}$

So, this vector field$F(x,y,z)=(y+z,2x+xz+2xy+y)$is not a conservative vector field

The divergence of a vector field $F(x,y,z)=F_1(x,y,z)i+F_2(x,y,z)j+F_3(x,y,z)k$is defined as follows

$divF=\nabla \times F$

Then, the divergence of the vector field $F(x,y,z)=(y+z,2x+xz+2xy+y)$will be,

$divF=\nabla \times F$

$=<\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}>.<y+z,2x+xz,2xy+y>$

$=0+0+0$

$=0$

So, the divergence for this non conservative vector field: $F=<y+z,2x+xz+2xy+y>$is uniformly equal to zero in ${\mathrm{ℝ}}^3$

Therefore an example of a non-conservative vector field whose divergence is uniformly equal to zero.