Continuum equations are ubiquitous in physical modeling of elastic, viscous, and viscoelastic systems. The equations of continuum mechanics take nontrivial forms on curved surfaces. Although the curved surface formulation of the continuum equations are derived in many excellent references available in the literature, they are not readily usable for solving physical problems due to the covariant, contravariant or mixed nature of the stress and strain tensors in the equations. We present the continuum equations in terms of physical components for general curved surfaces. This general formulation of the continuum equations can be used readily for modeling physical problems on arbitrary curved surfaces which appear frequently in biological cells and tissues as well as in engineering systems. We demonstrate this with the help of some examples.