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Chapter 10: Q6P (page 505)

Show that the sum of two 3rd-rank tensors is a 3rd-rank tensor. Hint: Write the transformation law for each tensor and then add your two equations. Divide out the factors to leave the result T'伪尾纬+S'伪尾纬=a伪ia尾ja纬k(Tijk+Sijk).

Short Answer

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Answer

The equation has been proven.

Step by step solution

01

Given Information

The two3rd-rank tensor.

02

Definition of a cartesian tensor.

The first rank tensor is just a vector. A tensor of the second rank has nine components (in three dimensions) in every rectangular coordinate system.

03

Prove the statement.

Let T and S be the two 3rd-rank tensor.

T+Sijk=Tijk+Sijk

Use transformation law in the above equation.

T+S'伪尾纬=a伪ia尾ia纬iTijk+a伪ia尾ia纬iSijkT+S'伪尾纬=a伪ia尾ia纬iTijk+a尾ia纬iSijkT+S'伪尾纬=a伪ia尾ia纬iTijk+a纬iSijkT+S'伪尾纬=a伪ia尾ia纬iTijk+Sijk

Hence, T+S'伪尾纬=a伪ia尾ia纬iT+Sijkand the sum of two 3rd-rank tensor is a -rank tensor.

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