  In this paper, we introduce a statistical field theory that describes the
macroscopic mechanical forces in inhomogeneous Coulomb fluids. Our approach
employs the generalization of Noether's first theorem for the case of
fluctuating order parameter, to calculate the stress tensor for Coulomb fluids.
This tensor encompasses the mean-field stress tensor and the fluctuation
corrections derived through the one-loop approximation. The correction for
fluctuations includes a term that accounts for the thermal fluctuations of the
local electrostatic potential and field in the vicinity of the mean-field
configuration. This correlation stress tensor determines how electrostatic
correlation affects local stresses in a nonuniform Coulomb fluid. We also use
previously formulated general covariant methodology [P.E. Brandyshev and Yu.A.
Budkov, J. Chem. Phys. 158, 174114 (2023)] in conjunction with a functional
Legendre transformation method and derive within it the same total stress
tensor. We would like to emphasize that our general approaches are applicable
not only to Coulomb fluids but also to nonionic simple or complex fluids, for
which the field-theoretic Hamiltonian is known as a functional of the relevant
scalar order parameters.
