## Christoffel Symbols

In mathematics and physics, the **Christoffel symbols**, named for Elwin Bruno Christoffel (1829–1900), are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. In a broader sense, the connection coefficients of an arbitrary (not necessarily metric) affine connection in a coordinate basis are also called Christoffel symbols. The Christoffel symbols may be used for performing practical calculations in differential geometry. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives.

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### Some articles on christoffel symbols:

**Christoffel Symbols**, Covariant Derivative

... In a smooth coordinate chart, the

**Christoffel symbols**of the first kind are given by and the

**Christoffel symbols**of the second kind by Here is the inverse matrix to the metric tensor ...

**Christoffel symbols**satisfy the symmetry relation which is equivalent to the torsion-freeness of the Levi-Civita connection ... The contracting relations on the

**Christoffel symbols**are given by and where

... of motion are expressed in a curvilinear coordinate system,

**Christoffel symbols**appear in the acceleration of a particle expressed in this coordinate system, as described below in more detail ... in general, in which the coefficients of the unit vectors are the

**Christoffel symbols**for the coordinate system ... The general notation and formulas for the

**Christoffel symbols**are and the

**symbol**is zero when all the indices are different ...

**Christoffel Symbols**- Applications To General Relativity

... The

**Christoffel symbols**find frequent use in Einstein's theory of general relativity, where spacetime is represented by a curved 4-dimensional Lorentz manifold with a Levi-Civita ... in the presence of matter—contain the Ricci tensor, and so calculating the

**Christoffel symbols**is essential ... by solving the geodesic equations in which the

**Christoffel symbols**explicitly appear ...

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