  Based on the Skyrme energy density functional, the spatial distribution of
the symmetry energy of a finite nucleus is derived in order to examine whether
the symmetry energy of a finite nucleus originates from its interior or from
its surface. It is found that the surface part of a heavy nucleus contributes
dominantly to its symmetry energy compared to its inner part. The symmetry
energy coefficient $a_{\text{sym}}({A})$ is then directly extracted and the
ratio of the surface symmetry coefficient to the volume symmetry coefficient
$\kappa$ is estimated. Meanwhile, with the help of experimental alpha decay
energies, a macroscopic method is developed to determine the symmetry energy
coefficient of heavy nuclei. The resultant $a_{\text{sym}}({A})$ is used to
analyze the density dependence of the symmetry energy coefficient of nuclear
matter around the saturation density, and furthermore, the neutron skin
thickness of $^{208}\text{Pb}$ is deduced which is consistent with the pygmy
dipole resonance analysis. In addition, it is shown that the ratio $\kappa$
obtained from the macroscopic method is in agreement with that from the Skyrme
energy density functional. Thus the two completely different approaches may
validate each other to achieve more compelling results.
