= Solution
In <piezoresponse force microscopy>, a conducting <atomic force microscopy> tip contacts a sample with a counter-electrode. Apply $V(t)=V_{\rm dc}+V_{\rm ac}\cos\omega t$. The <converse piezoelectric effect> produces an oscillatory displacement, approximately $u_\omega=d_{\rm eff}V_{\rm ac}$. Contact with the moving surface bends the cantilever, and a <lock-in amplifier> records the displacement amplitude and phase relative to the drive.
For opposite out-of-plane <ferroelectric domains>, the effective piezoelectric coefficient changes sign. Their vertical response therefore differs by approximately \b[$180^\circ$ in phase]. Equivalent antiparallel domains can have similar amplitudes, while cancellation or reduced response near a wall can produce an amplitude minimum. Phase identifies the relative response sign; an uncalibrated detector phase is not an absolute polarization direction.
Use a small drive below the switching threshold while imaging, a stable contact force and a well-characterized contact resonance. Electrostatic forces, contact stiffness and surface charging can also create signals at the drive frequency; observing contrast alone is insufficient evidence for polarization switching.<Imaging mechanism of piezoresponse force microscopy of ferroelectric surfaces by Sergei V. Kalinin and Dawn A. Bonnell>[2002, electromechanical and electrostatic contrast regimes]{ref}
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