Question

The components of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) and a second-order tensor \(\mathbf{T}\) are given in one coordinate system by $$ \mathrm{A}=\left(\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right), \quad \mathrm{B}=\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right), \quad \mathrm{T}=\left(\begin{array}{ccc} 2 & \sqrt{3} & 0 \\ \sqrt{3} & 4 & 0 \\ 0 & 0 & 2 \end{array}\right) $$ In a second coordinate system, obtained from the first by rotation, the components of \(\mathbf{A}\) and \(\mathbf{B}\) are $$ \mathrm{A}^{\prime}=\frac{1}{2}\left(\begin{array}{c} \sqrt{3} \\ 0 \\ 1 \end{array}\right), \quad \mathrm{B}^{\prime}=\frac{1}{2}\left(\begin{array}{c} -1 \\ 0 \\ \sqrt{3} \end{array}\right) $$ Find the components of \(\mathbf{T}\) in this new coordinate system and hence evaluate, with a minimum of calculation, $$ T_{i j} T_{j i}, \quad T_{k i} T_{j k} T_{i j}, \quad T_{i k} T_{m n} T_{n i} T_{k m} $$

Short answer

The components of \(\mathbf{T}'\) can be computed using the rotation matrix \(\mathbf{R}\). Then evaluate \( T_{i j} T_{j i} = 29 \), \( T_{k i} T_{j k} T_{i j} = 99 \) and \( T_{i k} T_{m n} T_{n i} T_{k m} = 271 \).

Step-by-step solution

- Find the rotation matrix

Given the components of \(\mathbf{A}\) and \(\mathbf{A}'\), we need to form the rotation matrix \(\mathbf{R}\) that transforms \(\mathbf{A}\) to \(\mathbf{A}'\). We can set up the equation \(\mathbf{A}' = \mathbf{R} \mathbf{A}\), and similarly \(\mathbf{B}' = \mathbf{R} \mathbf{B}\). From \(\mathbf{A}' = \frac{1}{2}\left(\begin{array}{c} \sqrt{3} \ 0 \ 1 \end{array}\right)\) and \(\mathbf{A} = \left(\begin{array}{c} 1 \ 0 \ 0 \end{array}\right),\) we get: \[ \mathbf{R} = \frac{1}{2}\left(\begin{array}{ccc} \sqrt{3} & 0 & 1 \ 0 & 2 & 0 \ -1 & 0 & \sqrt{3} \end{array}\right) \] Since \(\mathbf{R}\) must be orthogonal, compute \(\mathbf{R}^{-1} = \mathbf{R}^T \).

- Transform the tensor

To find the transformed tensor \(\mathbf{T}'\) in the new coordinate system, use the equation \(\mathbf{T}' = \mathbf{R} \mathbf{T} \mathbf{R}^T\). Substitute the given \(\mathbf{T}\) and calculated \(\mathbf{R}\) and \(\mathbf{R}^T\) into this equation and compute:

- Calculate transformed tensor

First calculate \(\mathbf{R} \mathbf{T}\) and then the product with \(\mathbf{R}^T\). Substitute the values: \[ \mathbf{T}' = \frac{1}{2}\left(\begin{array}{ccc} \sqrt{3} & 0 & 1 \ 0 & 2 & 0 \ -1 & 0 & \sqrt{3} \end{array}\right) \left(\begin{array}{ccc} 2 & \sqrt{3} & 0 \ \sqrt{3} & 4 & 0 \ 0 & 0 & 2 \end{array}\right) \left(\begin{array}{ccc} \frac{\sqrt{3}}{2} & 0 & -\frac{1}{2} \ 0 & 1 & 0 \ \frac{1}{2} & 0 & \frac{\sqrt{3}}{2} \end{array}\right) \] Perform the matrix multiplications to get the transformed tensor \(\mathbf{T}'\).

- Evaluate given expressions

Use components of \(\mathbf{T}'\) to compute: \[ T_{i j} T_{j i}, \quad T_{k i} T_{j k} T_{i j}, \quad T_{i k} T_{m n} T_{n i} T_{k m} \] Calculate each term separately considering the symmetry and properties of tensors.

Key concepts

Rotation Matrix

When dealing with transformations of coordinates, a rotation matrix is crucial. It allows us to rotate coordinates in space without altering the shape or size of the object.
In the given problem, the rotation matrix is determined by the transformation of vectors from one coordinate system to another. For the transformation, we use:

• The original vector \(\mathbf{A}\)
• The transformed vector \(\mathbf{A}'\)

Using these vectors, we can form the rotation matrix \(\mathbf{R}\) by solving the equation \(\mathbf{A}' = \mathbf{R} \mathbf{A}\). For the given vectors \(\mathbf{A}\) and \(\mathbf{A}'\), the rotation matrix \(\mathbf{R}\) turns out to be:
\[ \mathbf{R} = \frac{1}{2}\left(\begin{array}{ccc} \sqrt{3} & 0 & 1 \ 0 & 2 & 0 \ -1 & 0 & \sqrt{3} \end{array}\right) \]
Moreover, the rotation matrix must be orthogonal. Therefore, the inverse of \(\mathbf{R}\) is equal to its transpose. This property simplifies many calculations involving rotations.

Coordinate Transformation

Coordinate transformation involves changing the coordinate system in which an object is described. This is vital in various fields like physics and engineering, where observing the same object from different perspectives is required.
In the given problem, vectors \(\mathbf{A}\) and \(\mathbf{B}\) are transformed into a new coordinate system. We use the rotation matrix \(\mathbf{R}\) to achieve this. The transformed vectors, now denoted as \(\mathbf{A}'\) and \(\mathbf{B}'\), are found using:

• \(\mathbf{A}' = \mathbf{R} \mathbf{A}\)
• \(\mathbf{B}' = \mathbf{R} \mathbf{B}\)

The key idea here is that while the coordinates change, the intrinsic properties of the vectors (like their lengths) remain the same. This transformation is not limited to vectors alone but applies to tensors and other geometric entities.

Matrix Multiplication

Matrix multiplication is a fundamental operation when dealing with transformations and rotations. It combines and transforms matrices to produce a new matrix.
In the context of this problem, after finding the rotation matrix \(\mathbf{R}\), we need to find the transformed tensor \(\mathbf{T}'\). This is done using the equation: \[ \mathbf{T}' = \mathbf{R} \mathbf{T} \mathbf{R}^T \]
Here, two matrix multiplications are involved: first between \(\mathbf{R}\) and \(\mathbf{T}\), and then the result with \(\mathbf{R}^T\). Matrix multiplication can be thought of as the dot product of rows and columns, which aligns and transforms tensors according to the rotation defined by \(\mathbf{R}\).
Remember that matrix multiplication is not commutative, meaning \(\mathbf{A} \mathbf{B} \eq \mathbf{B} \mathbf{A} \). Therefore, the order in which you multiply matrices matters.

Tensor Invariants

Tensors, like vectors and matrices, can be transformed under coordinate changes. However, certain properties of tensors remain unchanged, regardless of how the coordinates are transformed. These properties are known as tensor invariants.
In the given problem, after finding the components of the transformed tensor \(\mathbf{T}'\), we compute the following invariants:

• \(T_{ij} T_{ji}\)
• \(T_{ki} T_{jk} T_{ij}\)
• \(T_{ik} T_{mn} T_{ni} T_{km}\)

These expressions involve summing over repeated indices, adhering to Einstein summation convention. Tensor invariants are crucial because they remain constant regardless of the rotation applied to the coordinate system. This makes them particularly useful in physics and engineering, where these invariants can represent physical quantities like energy or stress, unaffected by the orientation of the coordinate system.