Question

Show that the sum of two \(3^{\mathrm{rd}}\) -rank tensors is a \(3^{\mathrm{rd}}\) -rank tensor. Hint: Write the transformation law for each tensor and then add your two equations. Divide out the \(a\) factors to leave the result \(T_{\alpha \beta \gamma}^{\prime}+S_{\alpha \beta \gamma}^{\prime}=a_{\alpha i} a_{\beta j} a_{\gamma k}\left(T_{i j k}+S_{i j k}\right)\) using summation convention.

Short answer

The sum of two 3rd-rank tensors is another 3rd-rank tensor.

Step-by-step solution

- Write the Transformation Law for Each Tensor

Consider two 3rd-rank tensors, denoted as \(T_{ijk}\) and \(S_{ijk}\). Under a change of coordinates, the components of these tensors transform according to the law: \( T'_{\beta \beta \beta} = a_{\beta i} a_{\beta j} a_{\beta k} T_{ijk} \) for tensor \(T\), and similarly for tensor \(S\): \( S'_{\beta \beta \beta} = a_{\beta i} a_{\beta j} a_{\beta k} S_{ijk} \).

- Add the Transformation Laws

Add the transformation equations for \(T\) and \(S\) to get: \( T'_{\beta \beta \beta} + S'_{\beta \beta \beta} = a_{\beta i} a_{\beta j} a_{\beta k} T_{ijk} + a_{\beta i} a_{\beta j} a_{\beta k} S_{ijk} \).

- Factor out the Common Terms

Observe that the transformation matrix components \( a_{\beta i} a_{\beta j} a_{\beta k} \) can be factored out, leading to: \( T'_{\beta \beta \beta} + S'_{\beta \beta \beta} = a_{\beta i} a_{\beta j} a_{\beta k} (T_{ijk} + S_{ijk}) \).

- Conclude that the Sum Transforms as a Third-Rank Tensor

Since the right side of the equation \( a_{\beta i} a_{\beta j} a_{\beta k} (T_{ijk} + S_{ijk}) \) is in the form of a transformation law for a 3rd-rank tensor, it follows that the sum \( T_{ijk} + S_{ijk} \) transforms as a 3rd-rank tensor.

Key concepts

Third-rank tensors

Third-rank tensors are mathematical entities that extend beyond vectors and matrices, which are first and second-rank tensors, respectively. They are denoted with three indices, such as \( T_{ijk} \) or \( S_{ijk} \). These indices help specify the position of elements within the tensor.

Third-rank tensors arise in various fields such as physics, engineering, and computer science. They can represent complex multi-dimensional data or physical concepts like stress and strain in materials science.

Importantly, the rank of a tensor indicates the number of indices needed to uniquely define each entry in the tensor. For third-rank tensors, this means you need three indices, resulting in a three-dimensional array of numbers.

Tensor Addition

Adding tensors operates similarly to adding matrices or vectors, but care must be taken to match the structures. In our specific problem, we consider the tensors \( T_{ijk} \) and \( S_{ijk} \). To add them, simply perform element-wise addition, resulting in another tensor, denoted as \( T_{ijk} + S_{ijk} \).

This operation is straightforward because the sum of the corresponding elements (i.e., entries with the same indices) forms a new tensor. For instance, if \(T_{123}\) is an element in the tensor \(T_{ijk}\), and \(S_{123}\) is the matching element in \(S_{ijk}\), their sum will be \((T + S)_{123} = T_{123} + S_{123}\).

Consequently, tensor addition preserves the rank and structure of the resulting tensor. When adding two third-rank tensors, you still get a third-rank tensor.

Transformation Law

The transformation law specifies how tensor components change under a coordinate transformation. For a third-rank tensor \( T_{ijk} \), the transformation law is given by:

\[ T'_{\beta \beta \beta} = a_{\beta i} a_{\beta j} a_{\beta k} T_{ijk} \]

Here, \( a_{\beta i} \), \( a_{\beta j} \), and \( a_{\beta k} \) represent the transformation matrix components. The prime notation (e.g., \( T'_{\beta \beta \beta} \)) shows the tensor components in the new coordinate system.

When dealing with two third-rank tensors like \( T_{ijk} \) and \( S_{ijk} \), they transform as follows:

\[ T'_{\beta \beta \beta} = a_{\beta i} a_{\beta j} a_{\beta k} T_{ijk} \]
\[ S'_{\beta \beta \beta} = a_{\beta i} a_{\beta j} a_{\beta k} S_{ijk} \]

Adding the transformed tensors yields:

\[ T'_{\beta \beta \beta} + S'_{\beta \beta \beta} = a_{\beta i} a_{\beta j} a_{\beta k} T_{ijk} + a_{\beta i} a_{\beta j} a_{\beta k} S_{ijk} \]

This confirms the sum of two tensors also adheres to the transformation law for a third-rank tensor.

Summation Convention

Summation convention, also known as Einstein summation convention, is a shorthand used in tensor calculus to simplify expressions involving summations over repeated indices.

According to this convention, whenever an index appears twice in a single term (once in an upper position and once in a lower position), a sum over all possible values of that index is implied. For example:

\[ T'_{\beta \beta \beta} = a_{\beta i} a_{\beta j} a_{\beta k} T_{ijk} \]

Here, the indices \( i \), \( j \), and \( k \) are summed over, even though the summation signs are not explicitly shown. This notation reduces clutter and makes tensor equations more readable.

In our context, when we add the transformation laws of two third-rank tensors, the terms involving \( a_{\beta i} a_{\beta j} a_{\beta k} \) are summed automatically due to the summation convention. Hence, the expression:

\[ T'_{\beta \beta \beta} + S'_{\beta \beta \beta} = a_{\beta i} a_{\beta j} a_{\beta k} (T_{ijk} + S_{ijk}) \]

demonstrates the summation convention in practice, showing that tensor addition preserves the structure and transformation properties.