Let $F_i$ be a well behaved vector field in $\mathbb{R}^3$ which rapidly vanishes at infinity. (I am using here the index notation.) By the Helmholtz theorem, knowledge of divergence $\partial_i F_i$, and the curl $\partial_iF_j - \partial_j F_i$ of $F_i$ provides enough information to reconstruct the original vector field $F_i$.
I am wondering how is it that one can reconstruct $F_i$ only from divergence and curl, while the complete Jacobian $J_{ij} = \partial_j F_i$ corresponds additional information. Divergence is the trace of $J_{ij}$, curl is skew-symmetric part, while the symmetric part $\partial_j F_i + \partial_i F_j$, sometimes known as shear, does not participate in the Helmholtz theorem. I would like to intuitively understand this behavior.
In the case of a scalar function $f$, one needs to know all of its derivatives $\partial_i f$ in order to reconstruct $f$ via line integral. However, in the case of Helmholtz theorem, we are reconstructing $F_i$ as a volume integral. Does this make any difference?