Question

Show that \(\epsilon_{j_{1} j_{2} \ldots j_{N}} R_{i_{1}}^{j_{1}} \ldots R_{i_{N}}^{j_{N}}=\epsilon_{i_{1} i_{2} \ldots i_{N}} \operatorname{det} R\).

Short answer

The given relation is true due to the properties of the Levi-Civita symbol under permutations and the fact that the determinant of a rotational matrix is 1. We derived the same form on both sides by carefully changing the indices and using the Kronecker delta.

Step-by-step solution

Understanding Levi-Civita Symbol

The Levi-Civita symbol \(\epsilon_{j_{1} j_{2} \ldots j_{N}}\) behaves in a particular way under permutations (exchanges of indices). When two indices are identical, the Levi-Civita symbol is zero. If we swap a pair of indices, the symbol changes sign. This property will be instrumental for the steps to follow.

Applying the Properties of Levi-Civita Symbol

Consider the permutation \(i_{1} i_{2} \ldots i_{N}\) and index each \(i_{p}\) as \(i_{\sigma(p)}\), where \(\sigma\) is a permutation function. We have \(\epsilon_{j_{1} j_{2} \ldots j_{N}} R_{i_{1}}^{j_{1}} \ldots R_{i_{N}}^{j_{N}} = \epsilon_{j_{\sigma(1)} j_{\sigma(2)} \ldots j_{\sigma(N)}} R_{i_{\sigma(1)}}^{j_{\sigma(1)}} \ldots R_{i_{\sigma(N)}}^{j_{\sigma(N)}}\). By the permutation property of the Levi-Civita symbol, this equals \((-1)^{\sigma} \epsilon_{j_{1} j_{2} \ldots j_{N}} R_{i_{1}}^{j_{1}} \ldots R_{i_{N}}^{j_{N}}\).

Summation Over j's and Applying Kronecker Delta

Summing over j's we get \(\epsilon_{j_{1} j_{2} \ldots j_{N}} R_{i_{1}}^{j_{1}} \ldots R_{i_{N}}^{j_{N}} = \epsilon_{j_{\sigma(1)} j_{\sigma(2)} \ldots j_{\sigma(N)}} R_{i_{1}}^{j_{1}} \ldots R_{i_{N}}^{j_{N}} = (-1)^{\sigma} \delta_{i_{1} i_{2} \ldots i_{N}}\), where \(\delta\) is the Kronecker delta, which is 1 if and only if all indices are the same.

Final Comparison

Now, by comparing the right-hand side and the left-hand side under the summation over j's, it becomes obvious that \(\epsilon_{j_{1} j_{2} \ldots j_{N}} R_{i_{1}}^{j_{1}} \ldots R_{i_{N}}^{j_{N}} = \epsilon_{i_{1} i_{2} \ldots i_{N}} \operatorname{det} R\), because the determinant of the rotational matrix \(R\) is 1, and the Levi-Civita symbol leaves only one term.

Key concepts

Permutation Properties

The Levi-Civita symbol, denoted as \( \epsilon \), is deeply connected with the concept of permutations. A permutation can be thought of as a specific way to arrange or reorder a set of elements. More technically, it's a bijection from a set to itself, essentially a shuffling of elements without any addition or loss.

For example, if we have three elements labeled 1, 2, and 3, a permutation might be the order 2-3-1. In terms of permutation properties, the Levi-Civita symbol is zero if any two indices are the same, which corresponds to the concept of a permutation not allowing the repeat of an element.

Furthermore, the Levi-Civita symbol is sensitive to the order of indices — if we swap any two indices, the symbol changes its sign. This property mirrors the 'even' or 'odd' nature of a permutation, depending on the number of swaps needed to return to the original order. An even number of swaps results in a positive sign, while an odd number of swaps gives a negative sign. These properties are vital for understanding many tensor operations in physics and mathematics, particularly those involving cross products and determinants.

Kronecker Delta

The Kronecker delta, expressed as \( \delta \) in mathematical notation, functions as a mathematical 'identity element' for indices. It's defined such that \( \delta_{ij} \) is 1 if and only if \( i = j \), and is 0 otherwise.

For instance, \( \delta_{12} = 0 \) because 1 and 2 are different, while \( \delta_{33} = 1 \) since the indices match. The beauty of the Kronecker delta lies in its simplicity and utility. When summed over, it effectively 'picks out' terms from a sum where the indices match, allowing for the simplification of otherwise complex expressions. It plays a crucial role alongside the Levi-Civita symbol in tensor calculus and is commonly used in solving problems related to coordinates, unit vectors, and orthogonal transformations.

Rotation Matrix

A rotation matrix, often denoted as \( R \), is a square matrix that represents a rotation in a multi-dimensional space. For a rotation matrix to be valid, it must preserve the length (norm) of vectors upon transformation and must have a determinant equal to 1.

Rotation matrices are orthogonal, meaning that their inverse is equal to their transpose (\( R^{-1} = R^T \) for a valid rotation matrix \( R \)). This property is essential for maintaining the rigidity of transformations—imagine rotating an object without changing its shape or size. In practical terms, rotation matrices are fundamental to describing the rotation of objects in three-dimensional space, such as when considering the pose of a spacecraft or the orientation of a molecule.

Determinant

In linear algebra, the determinant is a scalar value that is a function of the entries of a square matrix. It provides important information about the matrix. Specifically, the determinant can tell you if a matrix has an inverse, if it is a transformation that preserves volume, or if the matrix will change the orientation of the space it transforms.

The determinant of a rotation matrix is of particular interest, as it will always be 1 or -1. A determinant of 1 indicates a proper rotation, maintaining the original orientation, while -1 indicates an improper rotation, including a reflection which reverses orientation. In physics, volume preservation and orientation are essential considerations in dynamics and kinematics, as well as in fields like electromagnetism and quantum mechanics.