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

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...

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

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

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

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

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

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...

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

\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...

rotation. The explicit form of a covariant transformation is best introduced with the transformation properties of the derivative of a function. Consider a scalar...

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

}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...

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

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

^{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...

}\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...

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

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

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

{\displaystyle \nabla } denotes the covariant derivative associated to Γ {\displaystyle \Gamma } . It is a covariant derivative along γ {\displaystyle \gamma...

notion of a directional derivative of a function from multivariable calculus is extended to the notion of a covariant derivative of a tensor. Many concepts...

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

is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation...

Hamiltonian variant of covariant classical field theory is the covariant Hamiltonian field theory where momenta correspond to derivatives of field variables...

interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, contraction, or...

Schäfer. The gauge covariant derivative Dμ is required to transform quark fields in manifest covariance; the partial derivatives that form the four-gradient...

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...

norm of the absolute value of a section of a line bundle to its covariant derivative. The diamagnetic inequality has an important physical interpretation...