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The gauge covariant derivative is like a generalization of the covariant derivative used in general relativity. If a theory has gauge transformations, it means that some physical properties of certain equations are preserved under those transformations. Likewise, the gauge covariant derivative is the ordinary derivative modified in such a way as to make it behave like a true vector operator, so that equations written using the covariant derivative preserve their physical properties under gauge transformations.

Fluid dynamics

In fluid dynamics, the gauge covariant derivative of a fluid may be defined as

$$\nabla_t \mathbf{v}:= \partial_t \mathbf{v} + (\mathbf{v} \cdot \nabla) \mathbf{v}$$

where $$\mathbf{v}$$ is a velocity vector field of a fluid.
Gauge theory

In gauge theory, which studies a particular class of fields which are of importance in quantum field theory, the minimally-coupled gauge covariant derivative is defined as

$$D_\mu := \partial_\mu - i e A_\mu$$

where A_\mu is the electromagnetic vector potential.
What happens to the covariant derivative under a gauge transformation

If a gauge transformation is given by

$$\psi \mapsto e^{i\Lambda} \psi$$

and for the gauge potential

$$A_\mu \mapsto A_\mu + {1 \over e} (\partial_\mu \Lambda)$$

then $$D_\mu$$ transforms as

$$D_\mu \mapsto \partial_\mu - i e A_\mu - i (\partial_\mu \Lambda) ,$$

and $$D_\mu \psi$$ transforms as

$$D_\mu \psi \mapsto e^{i \Lambda} D_\mu \psi$$

and $$\bar \psi := \psi^\dagger \gamma^0$$ transforms as

$$\bar \psi \mapsto \bar \psi e^{-i \Lambda}$$

so that

$$\bar \psi D_\mu \psi \mapsto \bar \psi D_\mu \psi$$

and $$\bar \psi D_\mu \psi$$ in the QED Lagrangian is therefore gauge invariant, and the gauge covariant derivative is thus named aptly.

On the other hand, the non-covariant derivative \partial_\mu would not preserve the Lagrangian's gauge symmetry, since

$$\bar \psi \partial_\mu \psi \mapsto \bar \psi \partial_\mu \psi + i \bar \psi (\partial_\mu \Lambda) \psi .$$

Quantum chromodynamics

In quantum chromodynamics, the gauge covariant derivative is [1]

$$D_\mu := \partial_\mu - i g \, A_\mu^\alpha \, \lambda_\alpha$$

where g is the coupling constant, A is the gluon gauge field, for eight different gluons \alpha=1 \dots 8, \psi is a four-component Dirac spinor, and where $$\lambda_\alpha$$ is one of the eight Gell-Mann matrices, $$\alpha=1 \dots 8$$.
General relativity

In general relativity, the gauge covariant derivative is defined as

$$\nabla_j v^i := \partial_j v^i + \Gamma^i {}_{j k} v^k$$

where $$\Gamma^i {}_{j k}$$ is the Christoffel symbol.