Index Notation

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I have a general question about index notation.

For an arbitrary quantity, a,

"a" denotes a scalar quantity.
"a_i" denotes a vector.
"a_ij" denotes a 2nd-order tensor.

So, if I have something like "a_i*e_ij*b_j"

Would this be like multiplying an nx1 vector, an mxm matrix, and an Lx1 vector? It would not be a possible operation, but I'm wondering if that what it means when you multiply quantities like that.
Indexing is usually used when you have components that are closely related, like entries in a matrix or sequence, but the indeces themselves do not imply any particular structure.

[itex](x_i)[/itex] could refer to a vector, sequence, n-tuple, or just a list of disparate objects.

Similarly [itex](a_{ij})[/itex] could be the entries of an m x n matrix or a doubly indexed sequence of sequences (commonly seen in diagonalizing proofs).

Provided the products are defined, one could have all sorts of indeces running around in a product.

You could be looking at Tensor notation:[/URL]
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That is known as the "Einstein summation convention" and should be denoting Cartesian Tensor Notation. Also if your subscripts are seperated by a comma, that implys differentiation.

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