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Tensor Rank


The total number of contravariant and covariant indices of a tensor. The rank R of a tensor is independent of the number of dimensions N of the underlying space.

An intuitive way to think of the rank of a tensor is as follows: First, consider intuitively that a tensor represents a physical entity which may be characterized by magnitude and multiple directions simultaneously (Fleisch 2012). Therefore, the number of simultaneous directions is denoted R and is called the rank of the tensor in question. In N-dimensional space, it follows that a rank-0 tensor (i.e., a scalar) can be represented by N^0=1 number since scalars represent quantities with magnitude and no direction; similarly, a rank-1 tensor (i.e., a vector) in N-dimensional space can be represented by N^1=N numbers and a general tensor by N^R numbers. From this perspective, a rank-2 tensor (one that requires N^2 numbers to describe) is equivalent, mathematically, to an N×N matrix.

rankobject
0scalar
1vector
2N×N matrix
>=3tensor

The above table gives the most common nomenclature associated to tensors of various rank. Some care must be exhibited, however, because the above nomenclature is hardly uniform across the literature. For example, some authors refer to tensors of rank 2 as dyads, a term used completely independently of the related term dyadic used to describe vector direct products (Kolecki 2002). Following such convention, authors also use the terms triad, tetrad, etc., to refer to tensors of rank 3, rank 4, etc.

Some authors refer to the rank of a tensor as its order or its degree. When defining tensors abstractly by way of tensor products, however, some authors exhibit great care to maintain the separation and distinction of these terms.


See also

Contravariant Tensor, Covariant Tensor, Dyad, Matrix, Scalar, Tensor, Tetrad, Triad, Vector

This entry contributed by Christopher Stover

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References

Fleisch, D. A Student's Guide to Vectors and Tensors. New York: Cambridge University Press, 2012.Kaliakin, V. N. "A Brief Review of Tensors." 2008. http://www.ce.udel.edu/faculty/kaliakin/appendix_tensors.pdf.Kolecki, J. C. "An Introduction to Tensors for Students of Physics and Engineering." 2002. http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf.Schnack, D. "Scalars, Vectors, Tensors, and Dyads." 2007. http://www.physics.wisc.edu/grads/courses/726-f07/files/Section_2_Vectors_06.pdf.

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Tensor Rank

Cite this as:

Stover, Christopher. "Tensor Rank." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/TensorRank.html

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