the three vector bosons to the weak isospin, whose components are written here as the Pauli matrices . Via the Higgs mechanism, these boson fields combine into the massless electromagnetic field and the fields for the three massive vector bosons and . General relativity The covariant derivative in general relativity is a special example of the gauge covariant derivative. It corresponds to the Levi Civita connection (a special Riemannian connection) on the tangent bundle (or the frame bundle) i.e it acts on tangent vector fields or more generally, tensors. It is usually written as instead of . In this special case, a choice of (local) coordinates not only gives partial derivatives , but they double as a frame of tangent vectors in which a vector field can be uniquely expressed as (this uses the definition of a vector field as an operator on smooth functions that satisfies a product rule i.e. a derivation). Therefore in this case "the internal indices are also space time indices". Upto slightly different normalisation (and notation) the gauge potential is the Christoffel symbol defined by . It gives the covariant derivative . The formal similarity with the gauge covariant derivative is more clear when the choice of coordinates is decoupled from the choice of frame of vector fields . Especially when the frame is orthonormal, such a frame is usually called a d-Bein. Then where . The direct analogue of the "gauge freedom" of the gauge covariant derivative is the arbitrariness of the choice of an orthonormal d-Bein at each point in space-time: local Lorentz invariance . However, in this case the more general independence of the choice of coordinates for the definition of the Levi Civita connection gives diffeomorphism or general coordinate invariance. Fluid dynamics In fluid dynamics, the gauge covariant derivative of a fluid may be defined as where is a velocity vector field of a fluid. See also Kinetic momentum Connection (mathematics) Minimal coupling Ricci calculus References Tsutomu Kambe, Gauge Principle For Ideal Fluids And Variational Principle. (PDF file.) Differential geometry Connection (mathematics) Gauge theories