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This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see: * Tensor * Tensor (intrinsic definition) * Application of tensor theory in engineering science For some history of the abstract theory see also Multilinear algebra. ==Classical notation== ;Ricci calculus The earliest foundation of tensor theory – tensor index notation. ;Tensor order The components of a tensor with respect to a basis is an indexed array. The ''order'' of a tensor is the number of indices needed. Some texts may refer to the tensor order using the term ''degree'' or ''rank''. ;Rank The rank of a tensor is the minimum number of rank-one tensor that must be summed to obtain the tensor. A rank-one tensor may be defined as expressible as the outer product of the number of nonzero vectors needed to obtain the correct order. ;Dyadic tensor A ''dyadic'' tensor is a tensor of order two, and may be represented as a square matrix. In contrast, a ''dyad'' is specifically a dyadic tensor of rank one. ;Einstein notation This notation is based on the understanding that in a term in an expression contains a repeated index letter, the default interpretation is that the product is summed over all permitted values of the index. For example if ''aij'' is a matrix, then under this convention ''aii'' is its trace. The Einstein convention is widely used in physics and engineering texts, to the extent that if summation is not to be applied, it is normal to note that explicitly. ;Kronecker delta ;Levi-Civita symbol ;Covariant tensor ;Contravariant tensor The classical interpretation is by components. For example in the differential form ''aidxi'' the components ''ai'' are a covariant vector. That means all indices are lower; contravariant means all indices are upper. ;Mixed tensor This refers to any tensor that has both lower and upper indices. Cartesian tensor Cartesian tensors are widely used in various branches of continuum mechanics, such as fluid mechanics and elasticity. In classical continuum mechanics, the space of interest is usually 3-dimensional Euclidean space, as is the tangent space at each point. If we restrict the local coordinates to be Cartesian coordinates with the same scale centered at the point of interest, the metric tensor is the Kronecker delta. This means that there is no need to distinguish covariant and contravariant components, and furthermore there is no need to distinguish tensors and tensor densities. All Cartesian-tensor indices are written as subscripts. Cartesian tensors achieve considerable computational simplification at the cost of generality and of some theoretical insight. ;Contraction of a tensor ;Raising and lowering indices ;Symmetric tensor ;Antisymmetric tensor ;Multiple cross products 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Glossary of tensor theory」の詳細全文を読む スポンサード リンク
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