<p>It has long been recognized that the theory of nonlinear elasticity provides a rich framework for a large variety of issues of interest to applied mathematicians. In particular, researchers with primary interest in nonlinear partial differential equations have been attracted to this area of continuum mechanics. However, the detailed theoretical background giving rise to the governing partial differential equations is not always familiar to non-specialists. The purpose of the present expository note is to attempt to alleviate this situation by describing a variety of nonlinear partial differential equations that have been found to govern the deformations of anti-plane shear and plane strain for isotropic incompressible hyperelastic solids in equilibrium.</p>