Tensor Rank

The total number of contravariant and covariant indices of a tensor. The rank  of a tensor is independent of the number of dimensions  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  and is called the rank of the tensor in question. In -dimensional space, it follows that a rank-0 tensor (i.e., a scalar) can be represented by  number since scalars represent quantities with magnitude and no direction; similarly, a rank-1 tensor (i.e., a vector) in -dimensional space can be represented by  numbers and a general tensor by  numbers. From this perspective, a rank-2 tensor (one that requires  numbers to describe) is equivalent, mathematically, to an  matrix.

rankobject0scalar1vector2 matrixtensor

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

https://mathworld.wolfram.com/TensorRank.html