Proofs involving covariant derivatives

proofProofs involving Christoffel symbols
This article contains proof of formulas in Riemannian geometry that involve the Christoffel symbols.wikipedia
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Christoffel symbols

Christoffel symbolChristoffel coefficientsChristoffel connection
This article contains proof of formulas in Riemannian geometry that involve the Christoffel symbols.

List of formulas in Riemannian geometry

formulas in Riemannian geometryreturn to article
This article contains proof of formulas in Riemannian geometry that involve the Christoffel symbols. : Q.E.D. (return to article) :The expression in parentheses is the Einstein tensor, so : Q.E.D. (return to article)
The "gradient" of the scalar curvature follows from the Bianchi identity (proof):

Curvature form

Bianchi identityBianchi identitiescurvature
Start with the Bianchi identity

Tensor contraction

contractioncontractingContract
Contract both sides of the above equation with a pair of metric tensors:

Metric tensor

metricmetricsround metric
Contract both sides of the above equation with a pair of metric tensors:

Q.E.D.

quod erat demonstrandumQED
: Q.E.D. (return to article) :The expression in parentheses is the Einstein tensor, so : Q.E.D. (return to article)

Kronecker delta

Kronecker delta functiongeneralized Kronecker deltadelta function
:where δ is the Kronecker delta.

Covariant derivative

covariant differentiationtensor derivativecovariant differential
:and since the covariant derivative of the metric tensor is zero (so it can be moved in or out of the scope of any such derivative), then

Einstein tensor

Einstein curvature tensor
:The expression in parentheses is the Einstein tensor, so : Q.E.D. (return to article)

Coordinate system

coordinatescoordinateaxis
Starting with the local coordinate formula for a covariant symmetric tensor field, the Lie derivative along a vector field is

Lie derivative

Lie bracketLie commutatorcommuting vector fields
Starting with the local coordinate formula for a covariant symmetric tensor field, the Lie derivative along a vector field is

Vector field

vector fieldsvectorgradient flow
Starting with the local coordinate formula for a covariant symmetric tensor field, the Lie derivative along a vector field is

Partial derivative

partial derivativespartial differentiationpartial differential
:here, the notation means taking the partial derivative with respect to the coordinate x^a.

Ricci calculus

tensor index notationabsolute differential calculusantisymmetrization of indices

Contracted Bianchi identities

A proof can be found in the entry Proofs involving covariant derivatives.