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Tensor Laplacian

The vector Laplacian can be generalized to yield the tensor Laplacian

A_(munu;lambda)^(;lambda)=(g^(lambdakappa)A_(munu;lambda))_(;kappa)
(1)
=g^(lambdakappa)(partial^2A_(munu))/(partialx^lambdapartialx^kappa)-g^(munu)Gamma^lambda_(munu)(partialA_(munu))/(partialx^lambda)
(2)
=1/(sqrt(g))partial/(partialx^nu)(sqrt(g)g^(munu)(partialA_(munu))/(partialx^mu))
(3)
=1/(sqrt(g))partial/(partialx^mu)(sqrt(g)g^(mukappa)(partialA_(munu))/(partialx^kappa))
(4)
=1/(sqrt(g))(sqrt(g)g^(mukappa)A_(munu,kappa))_(,mu),
(5)

where g_(;kappa) is a covariant derivative, g_(munu) is the metric tensor, g=det(g_(munu)), A_(munu,kappa) is the comma derivative (Arfken 1985, p. 165), and

Gamma^lambda_(munu)=1/2g^(kappalambda)((partialg_(mukappa))/(partialx^nu)+(partialg_(nukappa))/(partialx^mu)-(partialg_(munu))/(partialx^kappa))
(6)

is a Christoffel symbol of the second kind.

See also

Laplacian, Vector Laplacian

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 165-166, 1985.Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, 1988.

Referenced on Wolfram|Alpha

Tensor Laplacian

Cite this as:

Weisstein, Eric W. "Tensor Laplacian." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TensorLaplacian.html

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