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

the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first...

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

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

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

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...

+\omega \wedge \theta .} Equivalently, Θ = Dθ, where D is the exterior covariant derivative determined by the connection. The torsion form is a (horizontal)...

the exterior derivative gives the exterior algebra of differential forms on a manifold the structure of a differential graded algebra. The exterior derivative...

and a vector field, the Lie derivative of a differential form with respect to a vector field and the covariant derivative of a differential form with...

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

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

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

+\omega \wedge \theta =D\theta ,} where as above D denotes the exterior covariant derivative. The first Bianchi identity takes the form D Θ = Ω ∧ θ . {\displaystyle...

and a covariant vector is a list of numbers that transforms in the same way. Contravariant vectors are often just called vectors and covariant vectors...

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

for the other result. The interior product relates the exterior derivative and Lie derivative of differential forms by the Cartan formula (also known...

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

electromagnetic tensor, conventionally labelled F, is defined as the exterior derivative of the electromagnetic four-potential, A, a differential 1-form:...

Application of tensor theory in engineering Continuum mechanics Covariant derivative Curvature Diffusion tensor MRI Einstein field equations Fluid mechanics...

derivative is a one-form on the punctured plane. It is closed (its exterior derivative is zero) but not exact, meaning that it is not the derivative of...

tensor Operations Covariant derivative Exterior covariant derivative Exterior derivative Exterior product Hodge star operator Lie derivative Raising and lowering...

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

convention to compensate for the difficulty in describing contractions and covariant differentiation in modern abstract tensor notation, while preserving the...

\nu }=0.} The (contracted) Bianchi identities automatically ensure the covariant conservation of the stress–energy tensor in curved spacetimes: ∇ μ T μ...

) In general relativity, the partial derivatives used in special relativity are replaced by covariant derivatives. What this means is that the continuity...

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

mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It is therefore a tensor field of order three...

of covariant order 1 are the one-forms in V∗ (for this reason, the elements of the last two spaces are often called the contravariant and covariant vectors)...

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

}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...