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Chapter 10: Q3P (page 513)

Show that $δ_{i j} ϵ_{k l m}$ is an isotropic tensor of rank 5. Hint: Combine equations (5.4) and (5.7).

Short Answer

Expert verified

The expression $δ_{i j} \in_{k l m}$ is an isotropic tensor.

Step by step solution

01

Given Information

An isotropic tensor of rank 5is . $δ_{i j} \in_{k l m}$

02

Definition of Isomorphic tensor

Isotropic tensor is defined as a tensor with components that remain unaltered when the coordinate system is rotated arbitrarily.

03

Simplify the isotropic tensor

Use the result given in previous question.

$\begin{matrix} ϵ_{ρ λ σ}^{'} = a_{ρ i} a_{λ j} a_{σ k} ϵ_{i j k} \\ = ϵ_{ρ λ σ} \det A \\ = ϵ_{ρ λ σ} \end{matrix}$

It is known that $δ_{α β}^{'} = δ_{α β}$

Use the above result to get $δ_{α β}^{'} \in_{ρ λ σ}^{'} = δ_{α β} \in_{ρ λ σ}$

Hence proved that $δ_{i j} \in_{k l m}$ is an isotropic tensor.

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Most popular questions from this chapter

$V_{j}$ If role="math" localid="1659267226224" $V_{i} = g_{i j}, V^{j}$ is a contravariant vector and is a covariant vector, show that $U^{i} V_{j}$ is a $2^{n d}$ -rank mixed tensor. Hint:Write the transformation equations for U and V and multiply them.

Show that the contracted tensor $T_{i j k} V_{k}$ is a $2^{n d}$ -rank tensors.

Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions. In spherical coordinates $\nabla \times (r e_{θ}), \nabla (r c o s θ), \nabla . r$ .

Let be the tensor in . This is a -rank tensor and so has components. Most of the components are zero. Find the nonzero components and their values. Hint: See discussion after .

In equation $(10.13),$ let the variables be rectangular coordinates x, y, z, and let $x_{1}, x_{2}, x_{3}$ , be general curvilinear coordinates, orthogonal or not (see end of Section 8 ). Show that $J^{T} J$ is the $g_{ij}$ matrix in $(8.13)$ [or in $(8.16)$ for an orthogonal system]. Thus show that the volume element in a general coordinate system is $dV = \sqrt{g} {dx}_{1} {dx}_{2} {dx}_{3}$ where $g = {detg}_{ij}$ , and that for an orthogonal system, this becomes [by $(8.16)$ or $(10.19)$ ], $dV = h_{1} h_{2} h_{3} {dx}_{1} {dx}_{2} {dx}_{3}$ . Hint: To evaluate the products of partial derivatives in $J^{T} J$ , observe that the same expressions arise as in finding ${ds}^{2}$ . In fact, from $(8.11)$ and $(8.12)$ , you can show that row i times column j in $J^{T} J$ is just $a_{i} . a_{j} = g_{ij}$ in equations $(8.11)$ to $(8.14)$ .

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