Strontium ruthenate (Sr$_2$RuO$_4$) is a multiband superconductor that displays evidence of topological superconductivity, although a model of the order parameter that is consistent with all experiments remains elusive. We integrated a piezoelectric-based strain apparatus with a scanning superconducting quantum interference device (SQUID) microscope to map the diamagnetic response of single-crystal Sr$_2$RuO$_4$ as a function of temperature, uniaxial pressure, and position with micron-scale spatial resolution. We thereby obtained local measurements of the superconducting transition temperature $T_c$ vs. anisotropic strain $epsilon$ with sufficient sensitivity for comparison to theoretical models that assume a uniform $p_xpm ip_y$ order parameter. We found that $T_c$ varies with position and that the locally measured $T_c$ vs. $epsilon$ curves are quadratic ($T_cproptoepsilon^2$), as allowed by the C$_4$ symmetry of the crystal lattice. We did not observe the low-strain linear cusp ($T_cpropto left| epsilon right|$) that would be expected for a two-component order parameter such as $p_xpm ip_y$. These results provide new input for models of the order parameter of Sr$_2$RuO$_4$.
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