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We present a quantitative analysis of the response of a thin superconducting film to an applied nonuniform ac magnetic field. The analysis is directly applicable to the single-coil inductive measurement technique where a small coil driven by an ac current produces a nonuniform field, with Jz determined by the appearance of a third-harmonic component of the voltage generated across the coil. We derive a simple model to explain the response of the film to the applied magnetic field in the absence of weak links. This model is used to predict the third-harmonic voltage generated across the coil. The derivation of the model explains why superconducting films having thicknesses even less than the penetration depth screen out the nonuniform ac magnetic fields generated by the coil. A simplified version of the model yields analytic expressions that describe the magnitude and phase of the third-harmonic component at high drive currents while the full model yields excellent agreement with experimental measurements for our highest quality epitaxial films. In other films, the presence of weak links leads to a characteristic signature in the harmonic response at low drive currents. We have also found that ion irradiation can reduce the critical current densities by significant amounts without introducing weak links into the film.