Index Notation

Basic Rule

The full notation and array notation are very helpful when introducing the operations and rules in tensor analysis. However, tensor notation and index notation are more commonly used in the context of partial differential equations and tensor analysis.

In the index notation, indices are categorized into two groups: free indices and dummy indices. A free index means an "independent dimension" or an order of the tensor whereas a dummy index means summation. The following three basic rules must be met for the index notation.

1. The same index (subscript) may not appear more than twice in a product of two (or more) vectors or tensors. Thus \[A_{ik} u_{k} , A_{ik} B_{kj} , A_{ij} B_{jk} C_{nk} \] are valid, but \[A_{kk} u_{k} , A_{ik} B_{kk} , A_{ij} B_{ik} C_{ik} \] are meaningless.

2. Free indices on each term of an equation must agree. Thus \[u_{i} =v_{i} +w_{i} \] \[u_{i} =A_{ik} B_{kj} v_{j} +C_{ik} w_{k} \] are valid, but \[u_{i} =A_{ij} \] \[u_{j} =A_{ik} u_{k} \] \[u_{i} =A_{ik} v_{k} +w_{j} \] are meaningless.

3. Free and dummy indices may be changed without altering the meaning of an expression under the condition that Rules 1 and 2 are not violated. Thus, \[u_{j} =A_{jk} v_{k} \Leftrightarrow u_{i} =A_{ik} v_{k} \Leftrightarrow u_{j} =A_{ji} v_{i} .\]


Tensor Algebra in Index Notation

First, according to the meaning of free dimension, a vector $u$ and a second-order tensor $A$ should be written as $u_{i} $ and $ A_{ij} $, respectively. The free indices can be changed to other symbols. The basic operations in tensor algebra can be expressed using the index notation as follows.

  • Addition $u_{i} =v_{i} +w_{i} $$\Leftrightarrow $$u=v+w$
  • Dot Product $\lambda =u\cdot v$$\Leftrightarrow $$\lambda =u_{i} v_{i} $
  • Vector Product $u=v\times w$$\Leftrightarrow $$u_{i} =\varepsilon _{ijk} v_{j} w_{k} $
  • Dyadic Product $A=uv$$\Leftrightarrow $$A_{ij} =u_{i} v_{j} $
  • Addition $C=A+B$$\Leftrightarrow $$C_{ij} =A_{ij} +B_{ij} $
  • Transpose $A=B^{T} $$\Leftrightarrow $$A_{ij} =B_{ji} $
  • Scalar Products \[\lambda =A:B\Leftrightarrow \lambda =A_{ij} B_{ij}\] \[\lambda =A\cdot \cdot B\Leftrightarrow \lambda =A_{ij} B_{ji}\]
  • Inner Product of a Tensor and a Vector$u=Av$$\Leftrightarrow $ $u_{i} =A_{ij} v_{j} $
  • Inner Product of Two Tensors $C=A\cdot B$$\Leftrightarrow $ $C_{ij} =A_{ik} B_{kj} $

    Determinant \[\lambda =\det A\Leftrightarrow \lambda =\frac{1}{6} \varepsilon _{ijk} \varepsilon _{lmn} A_{li} A_{mj} A_{nk} =\varepsilon _{ijk} A_{l1} A_{j2} A_{k3} \] \[\Leftrightarrow \varepsilon _{lmn} \lambda =\varepsilon _{ijk} A_{li} A_{mj} A_{nk} =\varepsilon _{ijk} A_{il} A_{jm} A_{kn} \] Inverse $A_{ji}^{-1} =\frac{1}{2\det \left(A\right)} \varepsilon _{ipq} \varepsilon _{jkl} A_{pk} A_{ql} $

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