In physics, the gauge covariant derivative is a means of expressing how fields vary from place to place, in a way that respects how the coordinate systems used to describe a physical phenomenon can themselves change from place to place. The gauge covariant derivative is used in many areas of physics, including quantum field theory and fluid dynamics and in a very special way general relativity.
If a physical theory is independent of the choice of local frames, the group of local frame changes, the gauge transformations, act on the fields in the theory while leaving unchanged the physical content of the theory. Ordinary differentiation of field components is not invariant under such gauge transformations, because they depend on the local frame. However, when gauge transformations act on fields and the gauge covariant derivative simultaneously, they preserve properties of theories that do not depend on frame choice and hence are valid descriptions of physics. Like the covariant derivative used in general relativity (which is special case), the gauge covariant derivative is an expression for a connection in local coordinates after choosing a frame for the fields involved, often in the form of index notation.