  We present a general relation between the electromechanical couplings of
infinitesimal strain and electric field to arbitrary order, measured at fixed
voltage and at fixed electric field. We show that the improper response at
fixed field can be written as the strain derivative of the $n^{\text{th}}$
order susceptibility tensor, and the proper response at fixed voltage drop can
be written as the response at fixed field plus corrections for dilations and
90$^{\circ}$ rotations induced by strain. Our theory correctly reproduces the
proper piezoelectric response and we go beyond with the electrostrictive
response. We present first-principles calculations of the improper
electrostrictive response at fixed field, and illustrate how the correction is
used to obtain the proper response at fixed voltage. This distinction is of
high importance given the recent interest in giant electrostrictors exhibiting
electromechanical responses as large as the piezoelectric ones.
