Answer
The vector field $F$ is in-compressible.
Work Step by Step
When $F=A i+B j+C k$, then we have $div F=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}$
We are given that $F(x,y,z)=f(x) i+g(y) j+h(z) k$
Here, we have $div F= \nabla \cdot F=[\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}] \cdot [f(x) i+g(y) j+h(z) k] $
This implies that
$div F= \nabla \cdot F=\dfrac{\partial (i}{\partial x}+\dfrac{\partial (j)}{\partial y}+\dfrac{\partial (k)}{\partial z} \cdot (f(y,z) i+g(x,z) j+h(x,y) i=0$
Hence, we have the vector field $F$ is in-compressible.
