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...

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...

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

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

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

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...

+\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...

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

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

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

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

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...

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

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

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

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

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)...

neither strictly covariant nor strictly contravariant; at least one of the indices of a mixed tensor will be a subscript (covariant) and at least one...

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

is normal to note that explicitly. Kronecker delta Levi-Civita symbol Covariant tensor Contravariant tensor The classical interpretation is by components...

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

in multivariate calculus, and their manipulation is carried out using exterior algebra. Following Grassmann, developments in multilinear algebra were...

manifold Information geometry Fisher information metric covariant derivative exterior covariant derivative Levi-Civita connection parallel transport Development...

with covariant and contravariant vectors, where the position of an index indicates the type of vector, the first case usually applies; a covariant vector...

{\displaystyle \mathbf {b} } are vector-valued differentiable functions, then the derivative (denoted by a prime ′ {\displaystyle {}'} ) of a ⋅ b {\displaystyle \mathbf...