Give an example of a non-conservative vector field whose divergence is never equal to zero in ${\mathrm{ℝ}}^3$ .

The objective is to give an example of a non-conservative vector field whose divergence is never equal to zero in .

A vector field $F(x,y,z)=<F_1(x,y,z),F_2(x,y,z),F_3(x,y,z)>$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

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

For this vector field,

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

Notice that,

$$\begin{gathered} \frac{\partial F_3}{\partial y}=\frac{\partial }{\partial y}(2xy+z) \\ =2x \\ \frac{\partial F_2}{\partial z}=\frac{\partial }{\partial z}(2y+xz) \\ =x \end{gathered}$$

Hence in this case

$\frac{\partial F_3}{\partial y}\not\equiv \frac{\partial F_2}{\partial z}$

So, this vector field $F(x,y,z)=<x+y,2y+xz,2xy+z)$is a non 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

$divF=\nabla .F$

Then the divergence of vector field role="math" localid="1650792162611" $F(x,y,z)=<x+y,2y+xz,2xy+z>$will be

role="math" localid="1650792148741" $$\begin{gathered} divF=\nabla .F \\ =<\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}>.<x+y,2y+xz,2xy+z> \\ =\frac{\partial }{\partial x}(x+y)+\frac{\partial }{\partial y}(2y+xz)+\frac{\partial }{\partial z}(2xy+z) \\ =1+2+1 \\ =4 \\ \not\equiv 0 \end{gathered}$$

So, the divergence for this non-conservative vector field $F=(x+y,2y+xz,2xy+z)$is never equal to zero in ${\mathrm{ℝ}}^3$

Therefore, an example of a non-conservative vector field whose divergence is never equal to zero in is