  Using the Hugenholtz-Van Hove theorem, we derive general expressions for the
quadratic and quartic symmetry energies in terms of single-nucleon potentials
in isospin asymmetric nuclear matter. These analytical relations are useful for
gaining deeper insights into the microscopic origins of the uncertainties in
our knowledge on nuclear symmetry energies especially at supra-saturation
densities. As examples, the formalism is applied to two model single-nucleon
potentials that are widely used in transport model simulations of heavy-ion
reactions.
