# Transformation Properties of a tensor

• CAF123
In summary, the conversation discusses the properties of an array ##D_{ijk}## and its representation as a tensor. It is mentioned that if ##b_i = D_{ijk}a_{jk}## represents a vector, then the transformation properties under rotations of the coordinate axes of ##D_{ijk}## and ##D_{ijk} + D_{ikj}## can be determined using the rotation matrix ##L## and the quotient theorem for identifying tensors. Further calculations are shown to demonstrate the transformation properties of ##D_{ijk}## as a tensor.
CAF123
Gold Member

## Homework Statement

##D_{ijk}## is an array with ##3^3## elements, which is not known to represent a tensor. If for every symmetric tensor represented by ##a_{jk}## $$b_i = D_{ijk}a_{jk},$$ represents a vector, what can be said about the transformation properties under rotations of the coordinate axes of ##D_{ijk}## and ##D_{ijk} + D_{ikj}##?

## Homework Equations

Transformation between vector components in frame S' and components in frame S, where the rotation matrix ##L## takes ##S \rightarrow S'## is ##b_i' = l_{ij}b_j##

Quotient Theorem to identify tensors.

## The Attempt at a Solution

##D_{ijk}## may be represented as a matrix of size 3 x 3. Can I interpret the coordinate axes to mean, for example, the standard Cartesian axes?

$$b_i' = D_{ijk}' a_{jk}' = l_{ij}b_j = l_{ij}D_{jlm}a_{lm} = l_{ij} D_{jlm}a_{ml}\,\,\,\,(1)$$ using that ##a_{lm}## is a symmetric tensor.

Since ##a_{jk}## is a second rank tensor, it obeys the transformation law; ##a_{jk}' = l_{j\alpha}l_{k \beta} a_{\alpha \beta}##. Since the rotation matrix ##L## satisfies ##LL^{T} = I##, we have that ##l_{\beta k} l_{\alpha j}a_{jk}' = a_{\alpha \beta}## so then subbing into (1) gives$$l_{ij}D_{jlm} a_{ml} = l_{ij}D_{jlm} l_{lk}l_{mj}a_{jk}'$$
which is equal to $$l_{ij} (l_{jm})^{T} D_{jlm} l_{lk} a_{jk}' = D_{ijl}l_{lk} a_{jk}' = b_i'$$ using ##LL^T = I##. I am not really sure if this result helps or not. I think the aim of the question is to show using the transformation properties of a second rank symmetric tensor and a vector that ##D_{ijk}## is also a tensor, but I am not sure.

Can anyone give me a hint on how to progress?
Thanks.

## 1. What is a tensor?

A tensor is a mathematical object that describes the linear relationship between different sets of data. It is represented by an array of numbers, and its properties and transformations depend on its rank (number of dimensions) and its corresponding value in each dimension.

## 2. What are the transformation properties of a tensor?

The transformation properties of a tensor refer to how it changes when the coordinate system or basis vectors are transformed. This transformation can be represented by a matrix, and it allows us to understand how the tensor behaves in different reference frames.

## 3. How do you determine the rank of a tensor?

The rank of a tensor is determined by the number of indices needed to represent it. For example, a scalar (rank-0 tensor) is represented by a single number, a vector (rank-1 tensor) is represented by an array of numbers, and a matrix (rank-2 tensor) is represented by a 2D array of numbers.

## 4. What is the difference between contravariant and covariant tensors?

Contravariant and covariant tensors describe different types of transformations. Contravariant tensors are transformed by changing the basis vectors, while covariant tensors are transformed by changing the coordinate system. This means that contravariant tensors have their indices in the numerator of the transformation matrix, while covariant tensors have their indices in the denominator.

## 5. How are tensors used in science and engineering?

Tensors are used in a variety of scientific and engineering fields to model and understand complex systems. They are particularly useful in physics, engineering, and computer science for describing and analyzing physical phenomena, such as fluid dynamics, electromagnetism, and stress and strain in materials.

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