How do I see that $F$ is a vector field defined on all of $\mathbb{R}^3$? $$\vec{F}(x,y,z)= y^3z^3\mathbf{i} + 2xyz^3\mathbf{j} + 3xy^2z^2\mathbf{k}$$
How do I see that $F$ is a vector field defined on all of $\mathbb{R}^3$? And then is there an easy way to check if it has continuous partial derivatives?
I am looking at a theorem and it states, If F is a vector field defined on all of R^3 whose component functions have continuous partial derivatives and curl F = 0, then F is a conservative vector field. 
I don't know how to check those conditions. Could someone show with that problem given?

The vector field $F$ is defined on all of $\mathbb{R}^3$ because all of its component functions are (there are no pints where the functions are undefined, i.e., they make sense when plugging in any point of $\mathbb{R}^3$ into them). If say one of the component functions was $\frac{1}{x-y}$ then the vector field wouldn't be defined along the plane $x=y$. Moreover the component functions are continuous, as every polynomial function is continuous. To calculate the curl of the vector field just use the definition of curl, which involves just computing partial derivatives.
