Question

Write out the components of \(T_{j k}=A_{j} B_{k}-A_{k} B_{j}\) to show that \(T_{j k}\) is a \(2^{\text {nd }}\) -rank antisymmetric tensor with elements which are the components of \(\mathbf{A} \times \mathbf{B}\).

Short answer

\(T_{jk} = A_{j}B_{k} - A_{k}B_{j}\) is a 2nd-rank antisymmetric tensor with components matching \(\mathbf{A}\times\mathbf{B}\).

Step-by-step solution

Title - Expand the given tensor expression

Start by writing down the given tensor expression explicitly. The expression for the tensor is given as:\[ T_{j k} = A_{j} B_{k} - A_{k} B_{j} \]

Title - Verify antisymmetry

To show antisymmetry, demonstrate that \(T_{jk} = -T_{kj}\). Compute \(T_{kj}\) and compare it to \(T_{jk}\):\[ T_{k j} = A_{k} B_{j} - A_{j} B_{k} \]As seen, \(T_{kj} = -(A_{j} B_{k} - A_{k} B_{j}) = -T_{jk}\). Hence, \(T_{jk}\) is antisymmetric.

Title - Expand and match with cross product components

Identify the components of the tensor in three dimensions for \(j, k = 1, 2, 3\). The components correspond to the components of the cross product \(\mathbf{A} \times \mathbf{B}\). For each pair \( (j, k) \):\[ T_{12} = A_{1} B_{2} - A_{2} B_{1} \]\[ T_{13} = A_{1} B_{3} - A_{3} B_{1} \]\[ T_{23} = A_{2} B_{3} - A_{3} B_{2} \]These expressions are the components of the vector \(\mathbf{A} \times \mathbf{B}\):\[ (\mathbf{A} \times \mathbf{B})_{3} = A_{1} B_{2} - A_{2} B_{1} \]\[ (\mathbf{A} \times \mathbf{B})_{2} = -(A_{1} B_{3} - A_{3} B_{1}) \]\[ (\mathbf{A} \times \mathbf{B})_{1} = A_{2} B_{3} - A_{3} B_{2} \]

Title - Validate the tensor rank

Since \(T_{jk}\) involves two indices and transforms following coordinate changes, it is a 2nd-rank tensor. The antisymmetric nature and its components linking to the cross product confirm its rank and properties.

Key concepts

Antisymmetric Tensor

An antisymmetric tensor, also known as a skew-symmetric tensor, is a tensor that changes sign when its indices are swapped. In other words, for a tensor to be antisymmetric, we must have:

• T_{jk} = -T_{kj}

Consider the given tensor expression: \[ T_{jk} = A_j B_k - A_k B_j \] To verify the antisymmetric property, calculate \( T_{kj} \): \[ T_{kj} = A_k B_j - A_j B_k \] Notice that: \[ T_{kj} = - (A_j B_k - A_k B_j) = -T_{jk} \] This confirms that \( T_{jk} \) is indeed an antisymmetric tensor. Antisymmetric tensors are often found in physics and engineering, especially when dealing with rotational vectors and angular momentum.

Cross Product

The cross product, also known as the vector product, is an operation on two vectors in three-dimensional space. The result is a vector that is perpendicular to both of the original vectors. To find the cross product \( \textbf{A} \times \textbf{B} \) between vectors \( \textbf{A} = (A_1, A_2, A_3) \) and \( \textbf{B} = (B_1, B_2, B_3) \), we can use the following formula:

• \[ (\textbf{A} \times \textbf{B})_1 = A_2 B_3 - A_3 B_2 \]
• \[ (\textbf{A} \times \textbf{B})_2 = A_3 B_1 - A_1 B_3 \]
• \[ (\textbf{A} \times \textbf{B})_3 = A_1 B_2 - A_2 B_1 \]

The components of the cross product align with the components of the antisymmetric tensor \( T_{jk} \). For example, \( T_{12} = A_1 B_2 - A_2 B_1 \) is the same as \( (\textbf{A} \times \textbf{B})_3 \). Thus, the elements of an antisymmetric tensor can represent the components of the cross product.

Second-Rank Tensor

A second-rank tensor is an array of numbers arranged in a square matrix format, typically having two indices. These tensors describe linear relationships between two sets of vectors in various physical phenomena. The notation \( T_{jk} \) signifies that it has two indices: one row (j) and one column (k).To confirm that the given tensor is of the second rank, consider the tensor: \[ T_{jk} = A_j B_k - A_k B_j \]The presence of two indices explicitly (j and k) indicates it is indeed second rank. Second-rank tensors are essential in mechanics and physics as they represent stresses, strains, moments of inertia, and more.

Coordinate Transformation

Coordinate transformation is a mathematical tool used to convert the representation of vectors and tensors from one coordinate system to another. It ensures that the physical laws remain consistent regardless of the coordinate system used. For tensors, the transformation rules vary depending on their rank.For a second-rank tensor \( T_{jk} \), the transformation under a change of coordinates is given by: \[ T'_{jk} = \frac{\text{{∂} x^i}}{\text{{∂} x'^j}} \frac{\text{{∂} x^m}}{\text{{∂} x'^k}} T_{im} \]This transformation rule ensures that the form of the tensor remains consistent when shifting between reference frames. In the context of antisymmetric tensors, the transformation validates that the antisymmetric property is preserved across different coordinates.