mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way...

of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable principal...

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...

product rule. The Lie derivative is another derivative that is covariant under basis transformations. Like the exterior derivative, it does not depend on...

_{a}T_{c}} Another important tensorial derivative is the Lie derivative. Unlike the covariant derivative, the Lie derivative is independent of the metric, although...

calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to...

metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However...

The covariant derivative is a generalization of the directional derivative from vector calculus. As with the directional derivative, the covariant derivative...

\nabla y,} where ∇y is the covariant derivative of the tensor, and u(x, t) is the flow velocity. Generally the convective derivative of the field u·∇y, the...

notion of covariant derivative, because it is the monodromy of the ordinary differential equation on the curve defined by the covariant derivative with respect...

In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. The...

invariant of Riemannian metrics that measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only...

tensor fields: Lie derivatives, derivatives with respect to connections, the exterior derivative of totally antisymmetric covariant tensors, i.e. differential...

a covariant derivative ∇ in each associated vector bundle. If a local frame is chosen (a local basis of sections), then this covariant derivative is...

metric tensor g α β {\displaystyle g^{\alpha \beta }} and the tensor covariant derivative ∇ μ = ; μ {\displaystyle \nabla _{\mu }={}_{;\mu }} (not to be confused...

unique preferred torsion-free covariant derivative, known as the Levi-Civita connection. See also gauge covariant derivative for a treatment oriented to...

the scalar curvature, and ∇ ρ {\displaystyle \nabla _{\rho }} indicates covariant differentiation. These identities are named after Luigi Bianchi, although...

The expression ∇ X Y {\displaystyle \nabla _{X}Y} is called the covariant derivative of Y {\displaystyle Y} with respect to X {\displaystyle X} . Two...

determinant is formed by applying antisymmetrization to the indices. The covariant derivative ( ∇ {\displaystyle \nabla } ) is represented by a circle around the...

^{2}U^{i}}{\delta t^{2}}}\mathbf {e} _{i}\\\end{aligned}}} In terms of the covariant derivative, ∇ j {\displaystyle \nabla _{j}} , we have: δ U i δ t = V j ∇ j U...

a notion of Lie derivative on any associated bundle is obtained. If ∇ {\displaystyle \nabla } is a connection (or covariant derivative) on a vector bundle...

must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent...

group action. A covariant derivative in differential geometry is a linear differential operator which takes the directional derivative of a section of...

{\dot {\gamma }}} is the derivative with respect to t {\displaystyle t} . More precisely, in order to define the covariant derivative of γ ˙ {\displaystyle...

The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations...

the manifold. Therefore, we define the covariant derivative to be the forced projection of the intrinsic derivative back onto the manifold: a ⋅ D F = P B...

}\nabla _{a}e_{\nu }^{b}} where ∇ a {\displaystyle \nabla _{a}} is a covariant derivative, or equivalently a choice of connection on the frame bundle, most...

_{0}J^{\beta }} where the semicolon notation represents a covariant derivative, as opposed to a partial derivative. These equations are sometimes referred to as the...

Covariance and contravariance Covariant derivative Fictitious force Galilean invariance Gauge covariant derivative General covariant transformations Harmonic...

}f(\mathbf {p} )} (see Covariant derivative), L v f ( p ) {\displaystyle L_{\mathbf {v} }f(\mathbf {p} )} (see Lie derivative), or v p ( f ) {\displaystyle...