“Connection Coefficient”的概念、定义、习题解题方法及翻译-数学辞典

单词

Connection Coefficient

释义

Connection Coefficient

A quantity also known as a Christoffel Symbol of the Second Kind. Connection Coefficients are defined by

(1)

(long form) or

(2)

(abbreviated form), and satisfy

(3)

(long form) and

(4)

(abbreviated form).

Connection Coefficients are not Tensors, but have Tensor-like Contravariant and Covariant indices. A fully Covariantconnection Coefficient is given by

(5)

where the

s are the Metric Tensors, the

s are Commutation Coefficients, and the commas indicate the Comma Derivative. In an Orthonormal Basis,

and

, so

(6)

and

			(7)
			(8)
			(9)
			(10)
			(11)
			(12)

For Tensors of Rank 3, the connection Coefficients may beconcisely summarized in Matrix form:

(13)

Connection Coefficients arise in the computation of Geodesics. The GeodesicEquation of free motion is

(14)

(15)

Expanding,

(16)

(17)

But

(18)

(19)

where

(20)

See also Cartan Torsion Coefficient, Christoffel Symbol of the First Kind, Christoffel Symbol of theSecond Kind, Comma Derivative, Commutation Coefficient, Curvilinear Coordinates, SemicolonDerivative, Tensor

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