Understanding Index Notation: Multiplying Vectors & Tensors

In summary, index notation is used to represent different quantities, such as scalars, vectors, and tensors. When multiplying quantities in index notation, it is important to make sure that the products are defined. The convention for index notation is known as the "Einstein summation convention" and often denotes Cartesian Tensor Notation. Different subscripts can signify different operations, such as differentiation.
  • #1
womfalcs3
62
5
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.
 
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  • #2
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.

--Elucidus
 
  • #3
You could be looking at Tensor notation:

www.fm.vok.lth.se/Courses/MVK140/tensors.pdf[/URL]
 
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  • #4
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.
 

What is index notation?

Index notation is a mathematical notation system used to represent vectors and tensors. It uses subscripts to represent the components of a vector or tensor, making it easier to perform operations and calculations.

How do you multiply vectors using index notation?

To multiply two vectors using index notation, you simply multiply the corresponding components and then sum up the results. For example, to multiply two vectors A = (a1, a2, a3) and B = (b1, b2, b3), the resulting vector would be C = (a1b1 + a2b2 + a3b3).

What is the difference between multiplying vectors and multiplying tensors using index notation?

When multiplying tensors using index notation, you must also take into account the order of the tensors. This means that the resulting tensor will have a different number of components than the original tensors. In contrast, when multiplying vectors, the resulting vector will always have the same number of components as the original vectors.

Can you use index notation to multiply tensors of different orders?

Yes, you can use index notation to multiply tensors of different orders. However, the resulting tensor will have a different order than the original tensors. For example, multiplying a vector (1st-order tensor) and a matrix (2nd-order tensor) will result in a new tensor of 3rd order.

What are some common mistakes when using index notation for multiplying vectors and tensors?

Some common mistakes include not matching the dimensions of the tensors, not following the correct order of operations, and not properly labeling or tracking the indices. It is important to carefully check and double-check the indices and dimensions to ensure accurate results.

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