We study eigenvalue problems for the de Rham complex on varying three-dimensional domains. Our analysis includes the Helmholtz equation as well as the Maxwell system with mixed boundary conditions and non-constant coefficients. We provide Hadamard-type formulas for the shape derivatives under weak regularity assumptions on the domain and its perturbations. Our proofs are based on abstract results adapted to varying Hilbert complexes. As a byproduct of our analysis, we give a proof of the celebrated Hellmann–Feynman theorem both for simple and multiple eigenvalues of suitable families of self-adjoint operators in Hilbert space depending on possibly infinite dimensional parameters. This series of papers consists of Parts I and II.