In mathematics, the metric derivative is a notion of derivative appropriate to parametrized paths in metric spaces. It generalizes the notion of "speed"...

covariant derivative could be defined abstractly without the presence of a metric. The crucial feature was not a particular dependence on the metric, but that...

covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those geodesics of the metric that...

on [a,b]. For f ∈ ACp(I; X), the metric derivative of f exists for λ-almost all times in I, and the metric derivative is the smallest m ∈ Lp(I; R) such...

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

Riemannian metric in the case of Levi-Civita connection, or just an abstract connection) on the manifold. In contrast, when taking a Lie derivative, no additional...

relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. The metric captures all the...

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

covariant derivatives of the metric on E vanish. A principal connection on the bundle of orthonormal frames of E. A special case of a metric connection...

lengths of all such curves; this makes M a metric space. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable...

Y\rangle _{\gamma (s)}.} Taking the derivative at t = 0, the operator ∇ satisfies a product rule with respect to the metric, namely Z ⟨ X , Y ⟩ = ⟨ ∇ Z X ...

material derivative, including: advective derivative convective derivative derivative following the motion hydrodynamic derivative Lagrangian derivative particle...

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held...

Metric is a Canadian indie rock band founded in 1998 in Toronto, Ontario. The band consists of Emily Haines (lead vocals, synthesizers, guitar, tambourine...

in metric. Dieticians still use kilocalories, and doctors use millimetres of mercury. While these units are metric derivatives, they are not metric units...

Riemann, who first conceptualized them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent...

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

every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous. In the theory of differential equations, Lipschitz...

by the metric g. The relation between the exterior derivative and the gradient of a function on Rn is a special case of this in which the metric is the...

transported along it. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion. Geodesics are of particular importance in...

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

At this point the metric cannot be extended in a smooth manner (the Kretschmann invariant involves second derivatives of the metric), spacetime itself...

tensors. Over a Riemannian manifold, a metric (field of inner products) is available, and both metric and non-metric contractions are crucial to the theory...

In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein–Maxwell field equations, which corresponds to the gravitational...

important tensorial derivative is the Lie derivative. Unlike the covariant derivative, the Lie derivative is independent of the metric, although in general...

pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs...

is done horizontally, and matrix multiplication is done vertically. The metric tensor is represented by a U-shaped loop or an upside-down U-shaped loop...

the Fréchet derivative is a derivative defined on normed spaces. Named after Maurice Fréchet, it is commonly used to generalize the derivative of a real-valued...

derivative is another derivative that is covariant under basis transformations. Like the exterior derivative, it does not depend on either a metric tensor...

fields and exterior derivatives are introduced. A formal approach to the Minkowski metric A full-blown version of the Minkowski metric in coordinates as...