- #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?

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?