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Chapter 10: Q13P (page 505)

Show that the first parenthesis in (3.5) is a symmetric tensor and the second parenthesis is antisymmetric.

Short Answer

Expert verified

Answer

$T$ is a summation of an antisymmetric tensor and a symmetric tensor.

Step by step solution

01

Given information.

Physics definitions are given.

02

Definition of a tensor.

Tensors, like scalars and vectors, are mathematical constructs that can be used to describe physical qualities. Tensors are just a combination of scalars and vectors, with a scalar being a zero rank tensor and a vector being a first rank tensor.

03

Write the equation mentioned in the question.

Write the given equation.

$T_{i j} = \frac{1}{2} (T_{i j} + T_{j i}) + \frac{1}{2} (T_{i j} - T_{j i})$

Define new tensors.

$S_{i j} = \frac{1}{2} (T_{i j} + T_{j i}) A_{i j} = \frac{1}{2} (T_{i j} - T_{j i})$

Add the above equations.

$T_{i j} = S_{i j} + A_{i j}$

04

Invert the indices of the tensors.

Invert the indices of the tensors.

$S_{j i} = \frac{1}{2} (T_{j i} + T_{i j}) = S_{i j} A_{j i} = \frac{1}{2} (T_{j i} - T_{i j}) = - A_{i j}$

This implies that $T$ is a summation of an antisymmetric tensor and a symmetric tensor.

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Most popular questions from this chapter

In cylindrical coordinates ${鈭�}^{2} r, {鈭�}^{2} (\frac{1}{R}), {鈭�}^{2} l n_{r} .$

Parabolic cylinder coordinates $u, v, z .$

$x = u v c o s 蠒 y = u v s i n 蠒 z = \frac{1}{2} (u^{2} - v^{2})$

What are the physical components of the gradient in polar coordinates? [See (9.1)].The partial derivatives in (10.5) are the covariant components of $鈭� u$ . What relationdo you deduce between physical and covariant components? Answer the samequestions for spherical coordinates, and for an orthogonal coordinate system withscale factors $h_{1}, h_{2}, h_{3}$ .

$X_{i} A_{ij} = B_{j}$ .

For Example 1, write out the components of U,V, and $U 脳 V$ in the original right-handed coordinate system and in the left-handed coordinate system S' with the axis reflected. Show that each component of $U 脳 V$ inS'has the 鈥渨rong鈥� sign to obey the vector transformation laws.

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