Question

The third-rank tensor \(A_{i j k}\) is symmetric with respect to its first two suffices but anti-symmetric with respect to the second and third suffices. Show that all elements of \(A_{i j k}\) must be zero.

Short answer

Since the tensor is symmetric with respect to the first two indices and anti-symmetric with respect to the second and third indices, all elements of the tensor must satisfy \(A_{ijk} = -A_{ijk}\), which implies that \(A_{ijk} = 0\) for all i, j, k.

Step-by-step solution

Understanding Symmetry

For a tensor to be symmetric with respect to its first two indices, it means that exchanging these indices does not change the value of the tensor elements: so, for any indices i, j, k, we have that \(A_{ijk} = A_{jik}\).

Understanding Anti-symmetry

For a tensor to be anti-symmetric with respect to its second and third indices, it means that exchanging these indices flips the sign of the tensor elements: so, for any indices i, j, k, we have that \(A_{ijk} = -A_{ikj}\).

Applying Symmetry and Anti-symmetry

Combine the properties from Step 1 and Step 2, consider the element \(A_{ijk}\). By symmetry, \(A_{ijk} = A_{jik}\), and by anti-symmetry, \(A_{jik} = -A_{jki}\) which implies \(A_{ijk} = -A_{jki}\). As the tensor is symmetric in the first two indices, \(A_{ijk} = A_{jki}\) also holds. Therefore, we must have \(A_{ijk} = -A_{ijk}\), which is only possible if \(A_{ijk} = 0\) for all i, j, k.

Key concepts

Symmetric Tensors

Understanding symmetric tensors is fundamental in vector calculus and tensor analysis. A symmetric tensor is one that does not change its components when you swap two of its indices. For example, if we have a symmetric tensor denoted by \( A_{ij} \), then \( A_{ij} = A_{ji} \) for all values of indices \( i \) and \( j \). In other words, the elements on the opposite sides of the main diagonal are equal.

When dealing with higher-order tensors, symmetry can apply to more than just two indices. For instance, a third-rank tensor \( A_{ijk} \) might be symmetric with respect to its first two indices, which means \( A_{ijk} = A_{jik} \) for all combinations of \( i, j, \) and \( k \). This property simplifies calculations and has implications in physical laws, where symmetric tensors often represent conserved quantities or invariant properties under transformation.

Anti-Symmetric Tensors

In contrast to symmetric tensors, anti-symmetric (or skew-symmetric) tensors exhibit a change in sign when swapping certain indices. For a tensor \( B_{ij} \) to be anti-symmetric concerning indices \( i \) and \( j \), it must satisfy \( B_{ij} = -B_{ji} \). A tensor can be fully anti-symmetric, meaning this property holds for all pairs of its indices, or it can be anti-symmetric only with respect to specific pairs of indices.

An important consequence of anti-symmetry arises when we consider the diagonal elements of a matrix representation of a tensor. Since swapping the same index gives no change in position, the diagonal elements of an anti-symmetric tensor must be zero, as \( B_{ii} = -B_{ii} \). This property extends to higher dimensions, where a tensor can be anti-symmetric between different pairs of indices.

Tensor Properties

Tensors exhibit a set of vital properties that make them powerful tools for expressing physical laws and mathematical concepts. In addition to symmetry and anti-symmetry, tensors have other properties like linearity, meaning that tensor operations adhere to both the additive and multiplicative principles of scalar coefficients.

Moreover, tensors are characterized by their invariance under coordinate transformation, making them invaluable in physics where they describe quantities that are the same for all observers, irrespective of their reference frames. The tensor rank, often referred to as the order of the tensor, is an inherent property that indicates the number of indices or dimensions required to describe it. These indices are not just placeholders but represent directions in a multi-dimensional space, adding a layer of complexity and usefulness to the tensor concept.

Rank of a Tensor

The rank of a tensor provides critical information about its form and the complexity of the information it can convey. Mathematically, the rank (or order) of a tensor is defined by the number of indices needed to uniquely specify each of its components. Hence, a vector, which is a tensor of rank 1, requires one index, while a matrix, a tensor of rank 2, needs two indices.

For higher-order tensors, determining the rank gets more complex, with third-rank tensors needing three indices, and so on. The rank of a tensor tells us not only about the dimensions in which it operates but also about the potential relationships between different vectors and scalars encapsulated within it. It is crucial to note that zero elements do not necessarily diminish the rank of a tensor; instead, rank focuses on the maximum number of dimensions that define the tensor's structure.