Checking if a vector field is conservative I have three different vector fields and I want to check if they are conservative:
$$1)\space \space\vec{f}(\vec{x}):=\frac{1}{||\vec{x}||}\vec{a}, \space \space D=\mathbb R^2 / (\vec{0}), \space \space \vec{a}=(a_1,a_2)$$
$$2) \space \space \vec{g}(x,y)=(\frac{x}{y^2-x^2},\frac{-y}{y^2-x^2}), \space \space D=[(x,y) \in \mathbb R^2|\space y>|x|]$$
$$3) \space \space \vec{h}(x,y,z):=(2xz+y \cos (xy),\space x \cos(xy), \space x^2), D=[(x,y,z)\in \mathbb R^3 | \space x^2+y^2+z^2 \le 1]$$
I just learned/looked up the following on khanacademy:


*

*A vector field $\space \vec{f} \space $ is conservative $\iff$ $\underbrace{\int_{\gamma_2}\vec{v}(x) d\vec{x}=\int_{\gamma_1}\vec{v}(x) d\vec{x}}_{\text{Path independent}} \iff \underbrace{\oint_{\gamma}\vec{v}(x) d\vec{x}=0}_{\text{conservation of energy}}$

*If my vector field $\vec{v}$ is the gradient of some scalar field $V$ i.e $\vec{v}=\vec{\nabla}V \implies \vec{v}$ is conservative.
I understand what all of these things mean and how they were derived but I am somehow unable to properly apply them. Can someone maybe show me in a simple example how you would show that a vector field is conservative (doesn't have to be one of the fields I mentioned).  
Thanks in advance

EDIT: Answers for 2 and 3.
2. $\vec{g}(x,y)=(\frac{x}{y^2-x^2},\frac{-y}{y^2-x^2})$
If I have a vector field $\space f=P \hat{i}+Q \hat{j} \space$ then the field is conservative if $\frac{\partial P}{dy}=\frac{\partial Q}{dx}$
$$\frac{\partial P}{dy}=\frac{-2xy}{(y^2-x^2)^2}$$
$$\frac{\partial Q}{dx}=\frac{-2xy}{(y^2-x^2)^2}$$
$\implies$ the vector field is conservative
3. $\vec{h}(x,y,z)=(2xz,y \cos(xy),x \cos(xy),x^2)$
The vector field is conservative if: $\space \nabla \times \vec{h}=0$
$$\nabla \times \vec{h}=\begin{vmatrix}\hat{i}& \hat{j} & \hat{k}\\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}& \frac{\partial}{\partial z}\\ P&Q&R\end{vmatrix}=\hat{i}(0-0)-\hat{j}(2x-2x)+\hat{k}(\cos(xy)-xy\sin(xy)-\cos(xy)+xy \sin(xy))=0$$
$\implies$ vector field is conservative
