This book contains detailed insights on the calculus of variations that studies the equilibrium density matrix for many-particle Fermi systems. There are two approximations taken into account in the book. The simplest one is the mean field approximation. The second approximation applies not only to the particle distribution pattern but also to the correlation function. The variational principle for electron distribution function among wave vectors has been denoted in the work. The method that makes using the density matrix for finding the valence electron energy in metal has been proposed. It has been proved that the Coulomb interaction of electrons in a crystal lattice results in production of the model Hamiltonian consisting of two components. One component describes attraction of electrons with the equal wave vectors inducing formation of specific electron pairs. The other component describes repulsion of electrons with the wave vectors being equal by size and opposite by direction. This component denotes the anisotropy regarding distribution of electrons among wave vectors thus indicating the superconducting substance ability. The Fermi-Dirac distribution function, as considered together with the model Hamiltonian, explains all the properties exhibited by superconductors. Thus, the effect of a magnetic field on the superconducting states is studied in the book. The book can be interesting for senior schoolchildren, students of higher educational institutions, postgraduates and teachers.