Chapter 10: Q8P (page 534)

Using (10.15) show that $g_{i j}$ is a $2^{n d} V$ -rank covariant tensor. Hint:Write the transformationequation for each $d x$ , and set the scalar $d s'^{2} = d s^{2}$ to find the transformationequation for $g_{i j}$ .

Short Answer

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The results are proved in the solution.

Step by step solution

Given information.

The scalar $d s'^{2} = d s^{2}$

Definition of a 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 general form of a differential.

Length is an invariant quantity. This implies that $d s'^{2} = d s^{2}$ . Use $10.14, 10.15$ , and continue evaluation.

$\begin{matrix} d s^{2} = g_{i j} d x_{i} d x_{j} \\ d s^{2} = g_{i j} \frac{\partial x_{i}}{\partial x_{k}^{'}} \frac{\partial x_{j}}{\partial x_{l}^{'}} d x_{k}^{'} d x_{l}^{'} \\ d s^{' 2} = g_{k l}^{'} d x_{k}^{'} d x_{l}^{'} \end{matrix}$

Compare the equations and write the result.

Compare the deduced equations and write the result.

$g_{k l}^{'} = \frac{\partial x_{i}}{\partial x_{k}^{'}} \frac{\partial x_{j}}{\partial x_{l}^{'}} g_{i j}$

Thus, it can be concluded that $g_{i j}$ is a second order tensor.

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Using (10.15) show that $g_{i j}$ is a $2^{n d} V$ -rank covariant tensor. Hint:Write the transformationequation for each $d x$ , and set the scalar $d s'^{2} = d s^{2}$ to find the transformationequation for $g_{i j}$ .

Short Answer

Step by step solution

Given information.

Definition of a covariance and contravariance.

Write the general form of a differential.

Compare the equations and write the result.

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Using (10.15) show thatgijis a2ndV-rank covariant tensor. Hint:Write the transformationequation for eachdx, and set the scalards'2=ds2to find the transformationequation forgij.

Short Answer

Step by step solution

Given information.

Definition of a covariance and contravariance.

Write the general form of a differential.

Compare the equations and write the result.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity

Geometrical and Physical Optics

Energy Physics

Physics of Motion

Force

Mechanics and Materials

Study anywhere. Anytime. Across all devices.

Using (10.15) show that $g_{i j}$ is a $2^{n d} V$ -rank covariant tensor. Hint:Write the transformationequation for each $d x$ , and set the scalar $d s'^{2} = d s^{2}$ to find the transformationequation for $g_{i j}$ .