  Gauss integral theorems for electric and magnetic fields, Faradays law of
electromagnetic induction, magnetic field circulation theorem, theorems on the
flux and circulation of vector potential, which are valid in curved spacetime,
are presented in a covariant form. Covariant formulas for magnetic and electric
fluxes, for electromotive force and circulation of the vector potential are
provided. In particular, the electromotive force is expressed by a line
integral over a closed curve, while in the integral, in addition to the vortex
electric field strength, a determinant of the metric tensor also appears.
Similarly, the magnetic flux is expressed by a surface integral from the
product of magnetic field induction by the determinant of the metric tensor. A
new physical quantity is introduced - the integral scalar potential, the rate
of change of which over time determines the flux of vector potential through a
closed surface. It is shown that the commonly used four-dimensional
Kelvin-Stokes theorem does not allow one to deduce fully the integral laws of
the electromagnetic field and in the covariant notation requires the addition
of determinant of the metric tensor, besides the validity of the Kelvin-Stokes
theorem is limited to the cases when determinant of metric tensor and the
contour area are independent from time. This disadvantage is not present in the
approach that uses the divergence theorem and equation for the dual
electromagnetic field tensor. A new effect is predicted, according to which the
circulation of magnetic field can appear even in the absence of electric
current and with a constant electric field through the contour, if the area of
this contour would change. By analogy with electromagnetic induction, for the
magnetic field circulation to appear it is important that electric field flux
that passes through the area of the contour would change over time.
