How to prove being conservative vector field of $R^3$

Question

How to prove that $\mathbb{R}^3$ a conservative vector field $\mathbb{F}$ = (P, Q, R) satisfies these 3 conditions:

$P_y = Q_x$

$P_z = R_x$

$Q_z = R_y$

Answer 1

If $F = \nabla \varphi$, then $P = \varphi_x$, $Q = \varphi_y$ and $R=\varphi_z$. So $$P_y = (\varphi_x)_y=\varphi_{xy}=\varphi_{yx}=(\varphi_y)_x=Q_x.$$Same for the rest.

Answer 2

If a vector field is conservative, then it is the gradient of a function $f$. That is $$ F=\nabla f $$ for some function $f$. Since $\mathrm{curl} \nabla g=0$ regardless of the function $g$, you get that if $F$ is conservative then $$\mathrm{curl} F=\mathrm{curl} \nabla f=0\, . $$ This gives you the necessary conditions that $F$ has to satisfy and that you were looking for.