Question

(a) Show that for any general, but fixed, \(\phi\), $$ \left(u_{1}, u_{2}\right)=\left(x_{1} \cos \phi-x_{2} \sin \phi, x_{1} \sin \phi+x_{2} \cos \phi\right) $$ are the components of a first-order tensor in two dimensions. (b) Show that $$ \left(\begin{array}{cc} x_{2}^{2} & x_{1} x_{2} \\ x_{1} x_{2} & x_{1}^{2} \end{array}\right) $$ is not a (Cartesian) tensor of order 2. To establish that a single element does not transform correctly is sufficient.

Short answer

u_i transformations match first-order tensor rules. The given matrix fails to be a second-order tensor.

Step-by-step solution

Understand the transformation equations

Given transformation equations: \( u_{1} = x_{1} \cos\phi - x_{2} \sin\phi \) and \( u_{2} = x_{1} \sin\phi + x_{2} \cos\phi \)

Define first-order tensor transformation property

A first-order tensor transforms according to \(u_i' = T_{ij} x_j \), where \( T_{ij} \) is the transformation matrix and \( x_j \) is the original coordinate.

Write transformation matrix for rotation

Rotation transformation matrix \( T \) is \[ T = \begin{pmatrix} \cos\phi & -\sin\phi \ \sin\phi & \cos\phi \end{pmatrix} \]

Apply transformation matrix to original coordinates

Applying transformation matrix:\[ \begin{pmatrix} u_1 \ u_2 \end{pmatrix} = \begin{pmatrix} \cos\phi & -\sin\phi \ \sin\phi & \cos\phi \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \end{pmatrix}\]Calculate this to get \[ u_1 = x_1 \cos\phi - x_2 \sin\phi \]\[ u_2 = x_1 \sin\phi + x_2 \cos\phi \]

Verify first-order tensor condition

The results \( u_1 = x_1 \cos\phi - x_2 \sin\phi \) and \( u_2 = x_1 \sin\phi + x_2 \cos\phi \) match the transformation rules for a first-order tensor. Hence, it is proven.

Define second-order tensor transformation

A second-order tensor \( A \) transforms as \[ A_{ij}' = T_{ik} T_{jl} A_{kl} \], where \( A \) is the tensor and \( T \) is the rotation matrix.

Consider the given matrix

Given matrix: \[ A = \begin{pmatrix} x_2^2 & x_1 x_2 \ x_1 x_2 & x_1^2 \end{pmatrix} \]

Apply transformation

For element \(A_{11} \) after transformation: \[ A_{11}' = T_{1k} T_{1l} A_{kl} \] Calculate to check if it transforms correctly.

Check transformation for matrix element

Calculate \( A_{11} = x_2^2 \): \[ A_{11}' = \cos^2\phi \cdot x_2^2 + 2 \cos\phi \sin\phi \cdot (x_1 x_2) + \sin^2\phi \cdot x_1^2 \]

Identify mismatch

Upon calculation, element \( A_{11} \) will not yield \( x_2^2 \) back, proving it does not transform as a second-order tensor.

Key concepts

first-order tensor

A first-order tensor, also known as a vector, is a quantity that has both magnitude and direction. It transforms linearly when coordinates are rotated. Let's consider the transformation equations given:
\[ u_{1} = x_{1} \cos\text{\( phi \) } - x_{2} \sin\text{\( phi \) } \] and \[ u_{2} = x_{1} \sin\text{\( phi \) } + x_{2} \cos\text{\( phi \) } \].
These equations demonstrate the components of a first-order tensor in two dimensions. The components change according to the angle of rotation \( \phi \). As per the first-order tensor transformation property, we have \( u_{i}^{'} = T_{ij} x_{j} \), where \( T_{ij} \) is the transformation matrix and \( x_{j} \) is the original coordinate. By matching the results obtained after applying the rotation transformation matrix, we can verify that the given components indeed satisfy the first-order tensor transformation rules.

transformation matrix

A transformation matrix is a square matrix used to perform linear transformations such as rotations, scaling, and shearing. It provides a compact way to represent these transformations. For rotation transformations in two dimensions, the transformation matrix \( T \) is defined as:
\[ T = \begin{pmatrix} \cos\text{\( phi \) } & -\sin\text{\( phi \) } \ \sin\text{\( phi \) } & \cos\text{\( phi \) } \end{pmatrix} \]
By applying this matrix to the original coordinates \( x_{1} \) and \( x_{2} \), we obtain:
\[ \begin{pmatrix} u_{1} \ u_{2} \end{pmatrix} = \begin{pmatrix} \cos\text{\( phi \) } & -\sin\text{\( phi \) } \ \sin\text{\( phi \) } & \cos\text{\( phi \) } \end{pmatrix} \begin{pmatrix} x_{1} \ x_{2} \end{pmatrix} \]
After the multiplication, we get:
\( u_{1} = x_{1} \cos\text{\( phi \) } - x_{2} \sin\text{\( phi \) } \) and \( u_{2} = x_{1} \sin\text{\( phi \) } + x_{2} \cos\text{\( phi \) } \).
These results confirm that the vector transformation due to rotation is governed by the transformation matrix.

second-order tensor

A second-order tensor, also known as a matrix, is a mathematical object that can be represented as a 2D array of numbers. It transforms according to specific rules when the coordinate system is rotated. The general transformation rule for a second-order tensor \( A \) is given by:
\[ A_{ij}^{'} = T_{ik} T_{jl} A_{kl} \]
where \( T \) is the rotation matrix and \( A_{kl} \) are the elements of the tensor. In the given exercise, the matrix:
\[ A = \begin{pmatrix} x_{2}^{2} & x_{1} x_{2} \ x_{1} x_{2} & x_{1}^{2} \end{pmatrix} \]
is provided. By considering the transformation of the element \( A_{11} \), we need to calculate:
\( A_{11}^{'} = T_{1k} T_{1l} A_{kl} \).
After performing these calculations, it becomes evident that the transformed element \( A_{11} eq x_{2}^{2} \), indicating that the given matrix does not transform correctly according to the second-order tensor transformation rules.

rotation transformation

Rotation transformation involves rotating an object around a point or an axis. The amount of rotation is represented by the angle \( \phi \). In two-dimensional space, a point \( x \) with coordinates \( (x_{1}, x_{2}) \) can be rotated to a new point \( u \) with coordinates \( (u_{1}, u_{2}) \) using the rotation transformation matrix \( T \). The transformation equations to apply are:
\[ u_{1} = x_{1} \cos\text{\( phi \) } - x_{2} \sin\text{\( phi \) } \] and
\[ u_{2} = x_{1} \sin\text{\( phi \) } + x_{2} \cos\text{\( phi \) } \]
This rotation preserves the length of the vector, meaning it is an isometric transformation. Each new component is a combination of the sine and cosine of the rotation angle and the original coordinates. This ensures that every rotation aligns with the geometric interpretation of rotating a point around the origin by an angle \( \phi \).