For every closed path Γ, we have $\oint\limits_Γ (F_1 dx + F_2 dy + F_3 dz) = 0$.

There exists a differentiable scalar function $f(x, y, z)$ such that $F_1 = \frac{\partial f}{\partial x}$, $F_2 = \frac{\partial f}{\partial y}$, $F_3 = \frac{\partial f}{\partial z}$.

$\frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} = 0$.

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

$\begin{bmatrix}1 \\ -\sqrt{2/k}\end{bmatrix}$

$\begin{bmatrix}1 \\ \sqrt{2/k}\end{bmatrix}$

$\begin{bmatrix}\sqrt{2k} \\ 1\end{bmatrix}$

$\begin{bmatrix}\sqrt{2k} \\ -1\end{bmatrix}$