Vtyjkyuk

The gradient theorem states that if $F$ is a differentiable scalar-valued function of an $n$-dimensional vector space, then the gradient of $F$ is a conservative vector field. Line integrals through this field are path-independent, meaning that any line integral through $F'$ that starts at point $a$ and ends at point $b$ will evaluate to the scalar $F(b) - F(a)$.

Does this result generalize to cases where we start with a vector field? So say $G$ is a vector field and $G'$ is its Jacobian matrix: is the tensor field defined by $G'$ conservative in roughly the same way? Does any line integral through that field starting at $a$ and ending at $b$ now evaluate to the vector $G(b) - G(a)$?

I feel this must be true because you can just argue component-wise, treating each dimension of the output of $G$ as a scalar and considering the vector field defined by each of the corresponding dimensions of $G'$. But I'm not certain that actually works out, or whether there are additional conditions that must hold.

For example, must $G$ itself be a conservative vector field? Or, to put it differently, must $G'$ be a Hessian matrix?