TOPICS
Search

Covariant Derivative

The covariant derivative of a contravariant tensor A^a (also called the "semicolon derivative" since its symbol is a semicolon) is given by

A^a_(;b)=(partialA^a)/(partialx^b)+Gamma_(bk)^aA^k
(1)
=A^a_(,b)+Gamma_(bk)^aA^k
(2)

(Weinberg 1972, p. 103), where Gamma_(ij)^k is a Christoffel symbol of the second kind, Einstein summation is used in the last term, and A^a_(,b) is a comma derivative. The divergence del ·A=del _aA^a=A^a_(;a) is the tensor contraction of this covariant derivative, a scalar rather than the full (1,1) tensor (Carroll 1997, p. 61).

The covariant derivative of a covariant tensor A_a is

A_(a;b)=(partialA_a)/(partialx^b)-Gamma_(ab)^kA_k
(3)

(Weinberg 1972, p. 104).

Schmutzer (1968, p. 72) uses the older notation A^j_(∥k) or A_(j∥k).

See also

Christoffel Symbol, Comma Derivative, Covariant Tensor, Divergence, Levi-Civita Connection

Explore with Wolfram|Alpha

References

Carroll, S. M. "Lecture Notes on General Relativity." 3 Dec 1997. https://arxiv.org/abs/gr-qc/9712019. https://www.preposterousuniverse.com/wp-content/uploads/grnotes-three.pdf.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 48-50, 1953.Schmutzer, E. Relativistische Physik (Klassische Theorie). Leipzig, Germany: Akademische Verlagsgesellschaft, 1968.Weinberg, S. "Covariant Differentiation." §4.6 in Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, pp. 103-106, 1972.

Referenced on Wolfram|Alpha

Covariant Derivative

Cite this as:

Weisstein, Eric W. "Covariant Derivative." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CovariantDerivative.html

Subject classifications