# 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 demonstrandum∎QED**

: 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.

### Four-gradient

**4-gradientfour gradient**

### Ricci calculus

**tensor index notationabsolute differential calculusantisymmetrization of indices**

### Contracted Bianchi identities

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