Here are some important equations for General Relativity.

They are in Commentary Important Equations.pdf along with references and some notes.

Carroll's (1.68) where the Levi-Civita symbol is defined says "the Levi-Civita symbol is a ##(0,4)## tensor. It is NOT a tensor as he reminds us elsewhere.

They are in Commentary Important Equations.pdf along with references and some notes.

**Mathematics**

- Definition of ##{\partial }_{\mu }## (Carroll 1.54)
- (Anti)symmetrisation operator (Carroll 1.79)
- Vector as derivative (Carroll 2.16)
- Commutator (Carroll 2.20/2.23)
- Tensor transformation equation (Carroll 2.30)
- Basis vectors (Physics Forums)
- Covariant derivative / Christoffel symbol (Carroll section 3.2)
- Torsion Tensor (C Eq 3.22)
- The Christoffel connection ##\Gamma## (C Eq 3.27)
- The geodesic equation (C Eq 3.44)
- Directional covariant derivative (C Eq 3.38)
- The parallel transport equation (C Eq 3.39, 3.40)
- Riemann tensor (c Eq 3.112/3.113)
- Bianchi identity (c Eq 3.140)
- Ricci tensor and scalar, Weyl tensor (c Eq 3.144-3.147)
- Einstein tensor and 'contravariant' derivative (c Eq 3.1452,2)
- Killing's equation, Killing vectors (c Eq3.174)
- Geodesic deviation equation (c Eq3.208)

#### Tensor tricks

- Multi-dimensional Chain Rule
- Partial derivative of components gives Kronecker delta (C Eq 1.152)
- Partial derivatives commute
- Metric is always symmetric (C section 2.5)
- Contracting with metric lowers index
- Swap indices with metric
- A relationship for the derivative of the metric determinant
- Fully contracted symmetric × antisymmetric tensor vanishes
- Symmetrising a tensor equation
- Two formulas involving four-velocity
- The projection tensor
- Contra / co-variant tensor transformation matrices
- Tensor contractions using matrices

#### Physics

- Electro Magnetic Field Tensor (C Eq 1.69)
- Maxwell's equations (C Eq 1.96-1.98)
- Energy Momentum tensor for a perfect fluid (C Eq 3.93)
- Energy Momentum tensor for dust in SR (C Eq 1.110)
- Energy-momentum conservation equation (C Eq 3.92)

#### More Maths

- Differential and Integration on Web
- Pullback / Pushforward operators (Carroll A.9, A.10)
- Levi-Civita symbol and tensor (Carroll section 2.8)
- p-forms (Carroll section 2.9)
- Exterior derivative (Carroll 2.76)
- Wedge product (Carroll 2.73)
- Hodge star operator (Carroll 2.82)
- Stokes's theorem (C Eq 3.35)
- Euler-Lagrange Equation

All in Commentary Important Equations.pdf along with references and some notes.

Image from Wikipedia Babylonian equations.

Carroll's (1.68) where the Levi-Civita symbol is defined says "the Levi-Civita symbol is a ##(0,4)## tensor. It is NOT a tensor as he reminds us elsewhere.