# 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

The "gradient" of the scalar curvature follows from the Bianchi identity (proof):

### Curvature form

Bianchi identityBianchi identitiescurvature

### 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

### 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

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.