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Chapter 10: Q8P (page 517)

Write the transformation equations for $W 脳 S$ to verify the results of Example 3.

Short Answer

Expert verified

This answer proves that $W 脳 S$ is a polar vector.

Step by step solution

01

Given information.

Matrix definitions are given.

02

Definition of transformation laws.

Define the transformation laws.

${W_{i}}^{'} = a_{i j} W_{j} {S_{i}}^{'} = (d e t A) a_{i j} S_{j}$

03

Define the transformation laws.

Write the transformation laws assuming and are axial and polar vectors.

${W_{i}}^{'} = a_{ij} W_{j} {S_{i}}^{'} = (d e t A) a_{i j} S_{j}$

04

Continue the evaluations.

Write the implication of the transformation laws.

${(W' 脳 S')}_{伪} = 蔚'_{伪尾纬} W_{尾}' S_{纬}' = (d e t A) a_{伪 i} a_{尾 j} a_{纬 k} 蔚_{i j k} a_{尾 m} w_{m} (d e t A) a_{纬 m} S_{n} = {(d e t A)}^{2} a_{c i} 蔚_{i j k} w j S_{k} = a_{伪 i} {(W 脳 S)}_{I_{}}$

This implies that $W 脳 S$ is a polar vector.

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Most popular questions from this chapter

Show that in 2 dimensions (say the x , y plane), an inversion through the origin (that is $x' = - x, y' = - y$ ) is equivalent to a $180^{掳}$ rotation of the plane about the axis. Hint:Compare Chapter 3, equation (7.13) with the negative unit matrix.

$T = \frac{1}{2} m (蝇脳 r) 鈰� (蝇脳 r)$

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.

Point masses 1 at (1, 1, -2) and 2 at (1, 1, 1).

Verify equations(2.6).

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