I'm learning grad, div, and curl operators and a bit confused about the combinations of those.
So, I've learnt that $$\nabla \times (\nabla f)= \mathbf0$$ , and this implies that if $\nabla \times \mathbf F = \mathbf 0$ for some vector field $\mathbf F$, then $\mathbf F$ can always be written as a gradient of some scalar field $f$ .
I'm so confused about this implication. I know the other way of this implication is obvious (I mean the converse of this implication). But I don't know why this implication is true. Should the vector field $\mathbf F$ always have to be written as a gradient of some scalar field $f$ in this case? Is there any other way of writting $\mathbf F$ other than $\nabla f$ for some $f$?
Also, I've learnt that $$\nabla \cdot (\nabla \times \mathbf F) = 0$$ , and this implies that if $\nabla \cdot \mathbf G =0$ for some vector field $\mathbf G$, then $\mathbf G$ can be written as the curl of another vector field like, $\mathbf G = \nabla \times \mathbf F$. But this is one of the solutions. $\mathbf G$ can also be written as $\mathbf G = \nabla \times \mathbf G +\nabla f$ where $\nabla^2f = 0 $ and $\nabla \cdot \mathbf F = 0$.
I'm confused about this as well. Why can the vector field in the first one always be written as a gradient of some scalar field $f$? and why cannot this sense of implication apply to the vector field in the second one and so there are other solutions like $\mathbf G = \nabla \times \mathbf G +\nabla f$ as well?