How to prove ##V_{ai;j}=V_{aj;i}## in curved space using the given equation?

In summary: Also, how do you define raising/lowering indices?In summary, the first question asks if the identity involving the Levi-Civita tensor and the metric tensor holds in curved space, and if so, how to prove it. The second question asks for a way to show that ##V_{ai;j}=V_{aj;i}## is a possible solution to the given equation, which involves the Levi-Civita tensor and a vector field. The notation used in the conversation includes the Levi-Civita tensor ##\epsilon_{ijk}## and the metric tensor ##h_{ij}##, and there is a discussion about the raising and lowering of indices.
  • #1
user1139
72
8
Homework Statement
See below.
Relevant Equations
See below.
Question ##1##.

Consider the following identity

\begin{equation}
\epsilon^{ij}_{\phantom{ij}k}\epsilon_{i}^{\phantom{i}lm}=h^{jl}h^{m}_{\phantom{m}k}-h^{jm}h^{l}_{\phantom{l}k}
\end{equation}

which we know holds in flat space. Does this identity still hold in curved space? and if so, how does one go about proving it?

Question ##2##.

Consider the following

\begin{equation}
\epsilon^{ij}_{\phantom{ij}k}\left(V_{ai;j}-V_{aj;i}\right)=0.
\end{equation}

As ##\epsilon^{ij}_{\phantom{ij}k}## is not arbitrary, one cannot simply conclude that ##V_{ai;j}=V_{aj;i}##. Yet, I want to show that one can get ##V_{ai;j}=V_{aj;i}## from the above equation. Is there a way to do that rather than just saying that ##V_{ai;j}=V_{aj;i}## is a possible solution?
 
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  • #2
Please define your notation. What are ##\epsilon## and ##h##?
 
  • #3
##\epsilon_{ijk}## is the Levi-Civita tensor and ##h_{ij}## is the metric tensor in 3-space.
 
  • #4
Are you referring to the object with componens ##\pm 1## an 0 (which is not a tensor but a tensor density - and if so what do you mean by raising/lowering its indices) or to the actual tensor with components ##\pm 1## and 0 in an orthonormal coordinate system?
 

Related to How to prove ##V_{ai;j}=V_{aj;i}## in curved space using the given equation?

1. What are tensors and how are they used in science?

Tensors are mathematical objects that are used to describe the relationships between different physical quantities. They are commonly used in physics, engineering, and other scientific fields to model and analyze complex systems.

2. What is the difference between a tensor and a vector or matrix?

A vector is a one-dimensional tensor, while a matrix is a two-dimensional tensor. Tensors have more dimensions and can represent more complex relationships between quantities.

3. How are tensors represented and manipulated in mathematics?

Tensors are represented using indices, which indicate the dimensions and components of the tensor. They can be manipulated using tensor algebra, which involves operations such as addition, multiplication, and contraction.

4. What are some real-world applications of tensors?

Tensors are used in a wide range of fields, including physics, engineering, computer science, and data analysis. Some specific applications include analyzing stress and strain in materials, image and signal processing, and machine learning.

5. Are there different types of tensors?

Yes, there are several types of tensors, including scalars (zero-dimensional tensors), vectors (one-dimensional tensors), matrices (two-dimensional tensors), and higher-order tensors (three or more dimensions). Tensors can also be classified as covariant or contravariant, depending on how they transform under coordinate transformations.

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