Is $\vec{F} (x,y,z) = (x^2(y - z)+ yz)\vec{i} + (2xyz + x/z)\vec{j} + (y/x - 2xyz)\vec{k}$ solenoidal ?

Question

Check if div$\vec{F}$ = P′x+Q′y+R′z = or $\neq$ 0. Where $P = x^2(y - z)+ yz$, Q = 2xyz + x/z, R = y/x - 2xyz . Just as simple as that?

Answer 1

Yes, it is that simple.

In general if $\overrightarrow{F}(x,y,z) = P(x,y,z) \hat{x} + Q(x,y,z) \hat{y} + R(x,y,z) \hat{z}$, then

$\overrightarrow{F}(x,y,z)$ solenoidal $\iff$ div $ \displaystyle F = 0$

div $ \overrightarrow{F} = \left(\frac{\partial}{\partial x} \hat{x} + \frac{\partial}{\partial x} \hat{y} + \frac{\partial}{\partial z} \hat{z} \right) \cdot \left(P \hat{x} + Q \hat{y} + R \hat{z} \right) = \displaystyle\frac{\partial P}{\partial x} + \frac{\partial Q}{\partial x}+ \frac{\partial R}{\partial z} = 0$