Chapter 10: Q10P (page 534)

Show that if $V^{i}$ is a contravariant vector then $V^{i} = g_{i j} V^{j}$ is a covariant vector, andthat if $V^{i}$ is a covariant vector, then $V^{i} = g_{i j} V^{j}$ is a contravariant vector.

Short Answer

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Step by step solution

Given information.

A covariant tensor and a contravariant tensor are given.

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.

Begin by calculating the gradient.

Raise the index of $V_{i}^{'}$ since $V^{i}$ is contravariant

$\begin{matrix} V_{i}^{'} = g_{i j}^{'} V^{' j} \\ = g_{i j}^{'} \frac{\partial x_{j}^{'}}{\partial x_{k}} V^{k} \end{matrix}$

Continue evaluation.

Continue evaluation considering $g_{i j}$ is a second order covariant tensor.

$\begin{matrix} V_{i}^{'} = g_{l m} \frac{\partial x_{l}}{\partial x_{i}^{'}} \frac{\partial x_{m}}{\partial x_{j}^{'}} \frac{\partial x_{j}^{'}}{\partial x_{k}} V^{k} \\ δ_{k}^{m} = \frac{\partial x_{m}}{\partial x_{j}^{'}} \frac{\partial x_{j}^{'}}{\partial x_{k}} \\ V_{i}^{'} = \frac{\partial x_{l}}{\partial x_{i}^{'}} g_{l m} V^{m} \\ = \frac{\partial x_{l}}{\partial x_{i}^{'}} V_{l} \end{matrix}$

Recognize the covariant change in one brings a contravariant change in another.

$\begin{matrix} V'^{i} = g^{i j} V_{j}^{'} \\ = g^{i j} \frac{\partial x_{k}}{\partial x_{j}^{'}} V_{k} \\ g'^{i j} = g^{l m} \frac{\partial x_{i}^{'}}{\partial x_{l}} \frac{\partial x_{j}^{'}}{\partial x_{m}} \\ V^{' i} = g^{l m} \frac{\partial x_{i}^{'}}{\partial x_{l}} \frac{\partial x_{j}^{'}}{\partial x_{m}} \frac{\partial x_{k}}{\partial x_{j}^{'}} V_{k} \end{matrix}$

Continue the evaluation.

$\begin{matrix} δ_{m}^{k} = \frac{\partial x_{j}^{'}}{\partial x_{m}} \frac{\partial x_{k}}{\partial x_{j}^{'}} \\ V^{i} = \frac{\partial x_{i}^{'}}{\partial x_{l}} g^{l m} V_{m} \\ = \frac{\partial x_{i}^{'}}{\partial x_{l}} V^{l} \end{matrix}$

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Show that if $V^{i}$ is a contravariant vector then $V^{i} = g_{i j} V^{j}$ is a covariant vector, andthat if $V^{i}$ is a covariant vector, then $V^{i} = g_{i j} V^{j}$ is a contravariant vector.

Short Answer

Step by step solution

Given information.

Definition of a covariance and contravariance.

Begin by calculating the gradient.

Continue evaluation.

Recognize the covariant change in one brings a contravariant change in another.

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Show that ifViis a contravariant vector thenVi=gijVjis a covariant vector, andthat ifViis a covariant vector, thenVi=gijVjis a contravariant vector.

Short Answer

Step by step solution

Given information.

Definition of a covariance and contravariance.

Begin by calculating the gradient.

Continue evaluation.

Recognize the covariant change in one brings a contravariant change in another.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Space Physics

Force

Engineering Physics

Solid State Physics

Rotational Dynamics

Electrostatics

Study anywhere. Anytime. Across all devices.

Show that if $V^{i}$ is a contravariant vector then $V^{i} = g_{i j} V^{j}$ is a covariant vector, andthat if $V^{i}$ is a covariant vector, then $V^{i} = g_{i j} V^{j}$ is a contravariant vector.