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

Given information.

Physics definitions are given.

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.

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.

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

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

Short Answer

Step by step solution

Given information.

Definition of a tensor.

Write the equation mentioned in the question.

Invert the indices of the tensors.

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