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Chapter 10: Q12P (page 505)

XijklAkl=Bij.

Short Answer

Expert verified

Answer

The equation has been proven.

Step by step solution

01

Given Information

The tensorXijklAkl=Bij.

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 that X is a tensor.

Let XijklAkl=Bijwhere B is a non-zero tensor.

Apply transformation law on A and B.

0=X'伪尾蝇A'蝇-B'伪尾0=X'伪尾蝇A'蝇-a伪ia尾jBij0=X'伪尾蝇A'蝇-a伪ia尾jXijklAkl0=X'伪尾蝇A'蝇-a伪ia尾ja纬kanlXijklA'蝇0=X'伪尾蝇-a伪ia尾ja纬kanlXijklA'蝇

Let S伪=X'伪尾蝇-a伪ia尾ja纬kanlXijklA'伪尾

Hence, STA=0

The equation mentioned above is true for any matrix A.

Thus, X is a tensor.

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Most popular questions from this chapter

Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions. In spherical coordinates 鈭�脳(re胃),鈭�(rcos胃),鈭�.r.

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(b) Write equation(3.2)of Chapter 6 in tensor form to show the equivalence of the various expressions for the triple scalar product. Hint: Change the dummy indices as needed.

Use the results of Problem 1to find the velocity and acceleration components in spherical coordinates. Find the velocity in two ways: starting with ds and starting withs=rer.

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