Operation to make an (m+n)th rank tensor of rank-m and rank-n tensors

Instead, the Levi-Civita symbol can be represented as a 3D matrix with "layers" corresponding to rotation matrices and a single argument of theta = Pi/2. In summary, the conversation discusses the methods of creating rank-2 and rank-6 tensors from two rank-1 and rank-3 tensors, respectively. The possibility of representing this using a 6-dimensional matrix is considered, but ultimately deemed impractical. Instead, the Levi-Civita symbol can be represented as a 3D matrix with "layers" corresponding to rotation matrices and a single argument of theta = Pi/2.
  • #1
bjnartowt
284
3

Homework Statement



We know that c[ij] = ab[j] is a way to make a rank-2 tensor from two rank-1 tensors. We also know that C[abcxyz]=A[abc]B[xyz] is a way to make a rank-6 tensor from two rank-3 tensors. However, is there a matrix representation of this? I know the idea of a 6-dimensional matrix is painful and seemingly-unwieldy, but the question came up when I was writing myself some classical mechanics notes. The Levi-Civita symbol, if re-realized as a 3D "matrix", can be written as a 3D "matrix" whose "layers" are the three rotation matrices Rx, Ry, Rz, and feeding in the argument theta = Pi/2.


Homework Equations





The Attempt at a Solution

 
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  • #2
I don't think this is possible, as a 6-dimensional matrix would be far too unwieldy and difficult to work with.
 

FAQ: Operation to make an (m+n)th rank tensor of rank-m and rank-n tensors

What is an (m+n)th rank tensor?

An (m+n)th rank tensor is a mathematical object that represents a multidimensional array of values. It has m indices in one direction and n indices in another direction, making it a higher-order tensor.

How is an (m+n)th rank tensor formed?

An (m+n)th rank tensor is formed by combining two lower-order tensors of ranks m and n, respectively. This can be done through operations such as tensor multiplication or addition.

What is the purpose of creating an (m+n)th rank tensor?

The purpose of creating an (m+n)th rank tensor is to represent more complex data that cannot be adequately captured by lower-order tensors. It allows for a more comprehensive and efficient representation of multidimensional data.

How is an (m+n)th rank tensor used in scientific research?

An (m+n)th rank tensor is used in various fields of science, such as physics, engineering, and computer science. It is particularly useful in areas that involve multidimensional data, such as image and signal processing, fluid dynamics, and quantum mechanics.

Are there any limitations to creating an (m+n)th rank tensor?

The creation of an (m+n)th rank tensor is limited by the amount of available data and the computational resources needed to handle it. As the rank of the tensor increases, the amount of data and processing power required also increases, which can be a challenge in some cases.

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