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We propose a sensitivity analysis method that computes the S-parameter Jacobian from the volume E-field solutions in the frequency domain. The field solutions may be provided by any valid electromagnetic analysis. The computation is a post-process, which is independent from the simulator's grid, system equations, and discretization method. The sensitivity solver uses its own finite-difference grid and a sensitivity formula based on the finite-difference frequency-domain vector Helmholtz equation for the electric field. Its computational overhead is negligible in comparison with the simulation. We use the sensitivity solver in the gradient-based optimization of microwave structures. Significant reduction of the time required by the overall optimization process is achieved. In all design examples, the sensitivities are computed from the field solutions provided by a commercial finite-element simulator.