Understanding Index Notation: Multiplying Vectors & Tensors

[womfalcs3]

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.

 

[Elucidus]

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.

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

Similarly $(a_{ij})$ 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.

--Elucidus

 

[junglebeast]

You could be looking at Tensor notation:

www.fm.vok.lth.se/Courses/MVK140/tensors.pdf[/URL]

 

[CFDFEAGURU]

That is known as the "Einstein summation convention" and should be denoting Cartesian Tensor Notation. Also if your subscripts are separated by a comma, that implys differentiation.