Question

Write the equations in (2.16) and so in (2.17) solved for the unprimed components in terms of the primed components.

Short answer

Answer

The statement has been verified.

Step-by-step solution

Given Information

Vector U and vector V with components $U_1,U_2,U_3$and$V_1,V_2{V}_3$respectively.

Definition of a cartesian tensor.

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

Prove the statement.

Vector U and vector V with components $U_1,U_2,U_3$and $V_1,V_2,V_3$respectively.

A is an orthogonal matrix hence $A{A}^T={\delta }_{ij}$

The formula for the cartesian vectors states that $V_i=\sum _{j=1}^3{a}_{ij}{V}_j\text{'}$

$$\begin{gathered} V_i=\sum _{j=1}^3{a}_{ij}{V}_j\text{'} \\ {V}_i\left(U_i\right)=\sum _{j=1}^3{a}_{ij}{V}_j\text{'}\left(U_j\text{'}\right) \\ \sum _{kl}{a}_{ki}{a}_{ij}{V}_l\text{'}{U}_k\text{'}=\sum _k{a}_{ki}{U}_k\text{'}\sum _i{a}_{ji}{V}_j\text{'} \\ \sum _{kl}{a}_{ki}{a}_{ij}{V}_l\text{'}{U}_k\text{'}=U_i{V}_j \end{gathered}$$

Hence, the statement has been proven.