# Tensor analysis - Index Notation

Are there some good tutorials (.nb files) about Tensor analysis using index notation built in to Mathematica?

An example of a typical index notation:

$$C_{i j k l} = \lambda \delta_{i j} \delta_{k l} + \alpha \delta_{i k} \delta_{j l} + \beta \delta_{i l} \delta_{j k}$$

In Mathematica, symbolic tensors don't necessarily use indices. Instead, tensors are declared to have a certain number of indices (rank):

$Assumptions = A ∈ Arrays[{d, d}];  The above code asserts that A is a tensor of rank 2 where each index is of dimension d. For your example, let: id4 = TensorProduct[IdentityMatrix[d], IdentityMatrix[d]];  The tensor id4 corresponds to your$\lambda \delta_{i j} \delta_{k l}$term. Your other tensors are just transposed versions:$\alpha \delta_{i k} \delta_{j l}$: TensorTranspose[id4, {1,3,2,4}] $\beta \delta_{i l} \delta_{j k}$: TensorTranspose[id4, {1, 4, 2, 3}]  So, Mathematica would represent your tensor equation as: c = λ id4 + α TensorTranspose[id4, {1,3,2,4}] + β TensorTranspose[id4, {1,4,2,3}];  If you use an explicit dimension, then the above code would be represented using indices of an array. As an example, let the dimensions be$d=3\$, and check that the above equation sets the indices of c to the correct values:

d = 3;
c[[1, 2, 1, 2]]
c[[1, 1, 2, 2]]
c[[1, 2, 2, 1]]


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• thanking you! but in this case I only want to use index notation in mathematica, it should be helpful in some cases. Thanks you for your understanding! – ABCDEMMM Jul 2 '18 at 13:43
• Dear all. I only want to use build-in, not third part. If I search such topics in google, most are third part tool in mathematica. – ABCDEMMM Jul 2 '18 at 13:44
• @FEAPMAN TensorProduct and TensorTranspose are the built-in symbols that one can use with tensors. They are not from a third party package. – Carl Woll Jul 2 '18 at 13:48