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

单词

Covariant Tensor

释义

Covariant Tensor

A covariant tensor is a Tensor having specific transformation properties (c.f., a Contravariant Tensor). Toexamine the transformation properties of a covariant tensor, first consider the Gradient

(1)

for which

(2)

where

. Now let

(3)

then any set of quantities

which transform according to

(4)

or, defining

(5)

according to

(6)

is a covariant tensor. Covariant tensors are indicated with lowered indices, i.e.,

Contravariant Tensors are a type of Tensor with differing transformation properties,denoted . However, in 3-D Cartesian Coordinates,

(7)

for

, 2, 3, meaning that contravariant and covariant tensors are equivalent. The two types of tensors do differin higher dimensions, however. Covariant Four-Vectors satisfy

(8)

where

is a Lorentz Tensor.

To turn a Contravariant Tensor into a covariant tensor, use the Metric Tensor to write

(9)

Covariant and contravariant indices can be used simultaneously in a Mixed Tensor.

See also Contravariant Tensor, Four-Vector, Lorentz Tensor, Metric Tensor, Mixed Tensor,Tensor

References

Arfken, G. ``Noncartesian Tensors, Covariant Differentiation.'' §3.8 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 158-164, 1985.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 44-46, 1953.

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