Chapter 10: Q9P (page 534)

$V_{j}$ If role="math" localid="1659267226224" $V_{i} = g_{i j}, V^{j}$ is a contravariant vector and is a covariant vector, show that $U^{i} V_{j}$ is a $2^{n d}$ -rank mixed tensor. Hint:Write the transformation equations for U and V and multiply them.

Short Answer

Expert verified

The results are proved in the solution.

Step by step solution

Given information.

Co-variant and contra-variant components are defined.

Definition of covariance and contravariance.

The components of a vector relative to a tangent bundle basis are covariant in differential geometry if they change with the same linear transformation as the basis. If they change as a result of the inverse transformation, they are contravariant.

Write the definitions of the covariant and contravariant vectors.

Write the definitions of the vectors.

$U_{i}^{'} = \frac{\partial x_{i}^{'}}{\partial x_{k}} U^{k} V_{i}^{'} = \frac{\partial x_{j}}{\partial x_{l}^{'}} V_{l}$

Multiply the two equations.

Multiply the above equations.

$U_{i}^{'} V_{j}^{'} = \frac{\partial x_{i}^{'}}{\partial x_{k}} \frac{\partial x_{j}}{\partial x_{l}^{'}} U^{k} V_{l}$

It can be seen that this is a second-order mixed tensor.

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$V_{j}$ If role="math" localid="1659267226224" $V_{i} = g_{i j}, V^{j}$ is a contravariant vector and is a covariant vector, show that $U^{i} V_{j}$ is a $2^{n d}$ -rank mixed tensor. Hint:Write the transformation equations for U and V and multiply them.

Short Answer

Step by step solution

Given information.

Definition of covariance and contravariance.

Write the definitions of the covariant and contravariant vectors.

Multiply the two equations.

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VjIf role="math" localid="1659267226224" Vi=gij,Vjis a contravariant vector and is a covariant vector, show thatUiVjis a2nd -rank mixed tensor. Hint:Write the transformation equations for U and V and multiply them.

Short Answer

Step by step solution

Given information.

Definition of covariance and contravariance.

Write the definitions of the covariant and contravariant vectors.

Multiply the two equations.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Rotational Dynamics

Electricity and Magnetism

Solid State Physics

Fields in Physics

Medical Physics

Fundamentals of Physics

Study anywhere. Anytime. Across all devices.

$V_{j}$ If role="math" localid="1659267226224" $V_{i} = g_{i j}, V^{j}$ is a contravariant vector and is a covariant vector, show that $U^{i} V_{j}$ is a $2^{n d}$ -rank mixed tensor. Hint:Write the transformation equations for U and V and multiply them.