In the mathematical field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal bundle or vector bundle with a connection.

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.

In the math branches of differential geometry and vector calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to another two tangent vector fields.