In vector calculus, Helmholtz theorem says the divergence and curl of some vector field uniquely determines the vector field itself (with appropriate boundary conditions). Can this be generalized to differential forms?
That is, can a unique $p$-form $\alpha$ be determined from $d\alpha$ and $d\star\alpha$? Again assuming we're provided with appropriate boundary conditions. Here $d$ is the exterior derivative and $\star$ the Hodge dual.
Yes, such a generalization exists.
We have that $H^p_{{\rm dR}}(M)\cong \mathcal{H}^p(M,\mathtt{g})$ for every Riemannian metric $\mathtt{g}$ on $M$, where $\mathcal{H}^p(M,\mathtt{g})$ stands for the space of harmonic (i.e., closed and co-closed) $p$-forms on $M$. In other words, every cohomology class has a unique harmonic representative. Using this, one can show that every $p$-form $\omega$ on $M$ can be uniquely written as $\omega = {\rm d}\alpha + \delta \beta + \gamma$ for some $(p-1)$-form $\alpha$, $(p+1)$-form $\beta$, and harmonic $p$-form $\gamma$.
See https://en.wikipedia.org/wiki/Hodge_theory#Operators_in_Hodge_theory and the references therein for more details.