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

单词

Contravariant Tensor

释义

Contravariant Tensor

A contravariant tensor is a Tensor having specific transformation properties (c.f., a Covariant Tensor). Toexamine the transformation properties of a contravariant tensor, first consider a Tensor of Rank 1 (a Vector)

(1)

for which

(2)

Now let

, then any set of quantities

which transform according to

(3)

or, defining

(4)

according to

(5)

is a contravariant tensor. Contravariant tensors are indicated with raised indices, i.e.,

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

(6)

for

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

(7)

where

is a Lorentz Tensor.

To turn a Covariant Tensor into a contravariant tensor, use the Metric Tensor to write

(8)

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

See also Covariant 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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