Operator curl and gradient

Question

Operator curl $\nabla$ x$(\cdot)$ (x is cross product) working on a $ C^1$ vector field and operator gradient $\nabla(\cdot)$ working on scalar fields. And results of these operators is vector field. Is there any object (member) of kernel of curl in the images of gradient? (Any of members of kernel of curl is in the image of gradient is that true?) and is there any object (member) of images of gradient in the kernel of curl? (Any of members of image of gradient is in the kernel of curl is that true?)

Answer 1

$\text{curl}(\text{grad}\,f) = \mathbf 0$ for any function with continuous second-order partial derivatives. Provided we're in a simply connected region in space, then it is true that if $\text{curl}\,\mathbf F = \mathbf 0$, we have $F = \text{grad}\,f$. (Actually, something a bit weaker holds: You need the first cohomology $H^1(X,\mathbb R)$ to vanish; if $X$ is simply connected, this happens.)