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Qmechanic ♦
Qmechanic ♦
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I'm reading Carrol text on general relativity, on page 96 they arrive to the term $ \begin{equation} \frac{\partial x^{\mu}}{\partial x^{\mu '}}\frac{\partial x^{\lambda}}{\partial x^{\lambda '}}\frac{\partial^2 x^{\nu '}}{\partial x^{\mu}\partial x^{\lambda}}\tag{1} \end{equation}

\begin{equation} \frac{\partial x^{\mu}}{\partial x^{\mu '}}\frac{\partial x^{\lambda}}{\partial x^{\lambda '}}\frac{\partial^2 x^{\nu '}}{\partial x^{\mu}\partial x^{\lambda}}.\tag{1} \end{equation}

Can I contract this expression to get

\begin{equation} \frac{\partial^2 x^{\nu '}}{\partial x^{\mu '}\partial x^{\lambda '}}~?\tag{2} \end{equation}

I'm using thatthe chain rule $$\frac{\partial x^{\mu}}{\partial x^{\mu '}} \frac{1}{\partial x^{\mu}}=\frac{1}{\partial x^{\mu '}}\tag{3}$$$$\frac{\partial x^{\mu}}{\partial x^{\mu '}} \frac{\partial}{\partial x^{\mu}}=\frac{\partial}{\partial x^{\mu '}}\tag{3}$$ (which I think is correct).

I'm reading Carrol text on general relativity, on page 96 they arrive to the term $ \begin{equation} \frac{\partial x^{\mu}}{\partial x^{\mu '}}\frac{\partial x^{\lambda}}{\partial x^{\lambda '}}\frac{\partial^2 x^{\nu '}}{\partial x^{\mu}\partial x^{\lambda}}\tag{1} \end{equation}

Can I contract this expression to get

\begin{equation} \frac{\partial^2 x^{\nu '}}{\partial x^{\mu '}\partial x^{\lambda '}}~?\tag{2} \end{equation}

I'm using that $$\frac{\partial x^{\mu}}{\partial x^{\mu '}} \frac{1}{\partial x^{\mu}}=\frac{1}{\partial x^{\mu '}}\tag{3}$$ (which I think is correct).

I'm reading Carrol text on general relativity, on page 96 they arrive to the term

\begin{equation} \frac{\partial x^{\mu}}{\partial x^{\mu '}}\frac{\partial x^{\lambda}}{\partial x^{\lambda '}}\frac{\partial^2 x^{\nu '}}{\partial x^{\mu}\partial x^{\lambda}}.\tag{1} \end{equation}

Can I contract this expression to get

\begin{equation} \frac{\partial^2 x^{\nu '}}{\partial x^{\mu '}\partial x^{\lambda '}}~?\tag{2} \end{equation}

I'm using the chain rule $$\frac{\partial x^{\mu}}{\partial x^{\mu '}} \frac{\partial}{\partial x^{\mu}}=\frac{\partial}{\partial x^{\mu '}}\tag{3}$$ (which I think is correct).

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Qmechanic ♦
Qmechanic ♦
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I'm reading Carrol text on general relativity, on page 96 they arrive to the term $ \begin{equation} \frac{\partial x^{\mu}}{\partial x^{\mu '}}\frac{\partial x^{\lambda}}{\partial x^{\lambda '}}\frac{\partial^2 x^{\nu '}}{\partial x^{\mu}\partial x^{\lambda}}\tag{1} \end{equation}

Can I contract this expression to get

\begin{equation} \frac{\partial^2 x^{\nu '}}{\partial x^{\mu '}\partial x^{\lambda '}} \end{equation}\begin{equation} \frac{\partial^2 x^{\nu '}}{\partial x^{\mu '}\partial x^{\lambda '}}~?\tag{2} \end{equation}

I'm using that $\frac{\partial x^{\mu}}{\partial x^{\mu '}} \frac{1}{\partial x^{\mu}}=\frac{1}{\partial x^{\mu '}}$$$\frac{\partial x^{\mu}}{\partial x^{\mu '}} \frac{1}{\partial x^{\mu}}=\frac{1}{\partial x^{\mu '}}\tag{3}$$ (which I think is correct).

I'm reading Carrol text on general relativity, on page 96 they arrive to the term $ \begin{equation} \frac{\partial x^{\mu}}{\partial x^{\mu '}}\frac{\partial x^{\lambda}}{\partial x^{\lambda '}}\frac{\partial^2 x^{\nu '}}{\partial x^{\mu}\partial x^{\lambda}} \end{equation}

Can I contract this expression to get

\begin{equation} \frac{\partial^2 x^{\nu '}}{\partial x^{\mu '}\partial x^{\lambda '}} \end{equation}

I'm using that $\frac{\partial x^{\mu}}{\partial x^{\mu '}} \frac{1}{\partial x^{\mu}}=\frac{1}{\partial x^{\mu '}}$ (which I think is correct)

I'm reading Carrol text on general relativity, on page 96 they arrive to the term $ \begin{equation} \frac{\partial x^{\mu}}{\partial x^{\mu '}}\frac{\partial x^{\lambda}}{\partial x^{\lambda '}}\frac{\partial^2 x^{\nu '}}{\partial x^{\mu}\partial x^{\lambda}}\tag{1} \end{equation}

Can I contract this expression to get

\begin{equation} \frac{\partial^2 x^{\nu '}}{\partial x^{\mu '}\partial x^{\lambda '}}~?\tag{2} \end{equation}

I'm using that $$\frac{\partial x^{\mu}}{\partial x^{\mu '}} \frac{1}{\partial x^{\mu}}=\frac{1}{\partial x^{\mu '}}\tag{3}$$ (which I think is correct).

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Juan Pablo Arcila
Juan Pablo Arcila
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Can I contract index in this expression?

I'm reading Carrol text on general relativity, on page 96 they arrive to the term $ \begin{equation} \frac{\partial x^{\mu}}{\partial x^{\mu '}}\frac{\partial x^{\lambda}}{\partial x^{\lambda '}}\frac{\partial^2 x^{\nu '}}{\partial x^{\mu}\partial x^{\lambda}} \end{equation}

Can I contract this expression to get

\begin{equation} \frac{\partial^2 x^{\nu '}}{\partial x^{\mu '}\partial x^{\lambda '}} \end{equation}

I'm using that $\frac{\partial x^{\mu}}{\partial x^{\mu '}} \frac{1}{\partial x^{\mu}}=\frac{1}{\partial x^{\mu '}}$ (which I think is correct)