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Níckolas Alves
Níckolas Alves
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It may seem like a dumb question but I'm trying to solve a problem involving coordinate transformations on the Christoffel symbol and to solve it they do the following product rule:

  $$\partial_\alpha g_{\beta ' \gamma '} = \frac{\partial}{\partial x^{\alpha}} \left(\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \right)g_{\beta \gamma} + \frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}.$$

I understand that we need to take the product rule, but I don't understand why the second term is $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}$ and not $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial}{\partial x^{\alpha }} g_{\beta \gamma}$. Where does the $\frac{\partial x^{\nu}}{\partial x^{\alpha }}$ come from?

It may seem like a dumb question but I'm trying to solve a problem involving coordinate transformations on the Christoffel symbol and to solve it they do the following product rule:

$$\partial_\alpha g_{\beta ' \gamma '} = \frac{\partial}{\partial x^{\alpha}} \left(\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \right)g_{\beta \gamma} + \frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}.$$

I understand that we need to take the product rule, but I don't understand why the second term is $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}$ and not $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial}{\partial x^{\alpha }} g_{\beta \gamma}$. Where does the $\frac{\partial x^{\nu}}{\partial x^{\alpha }}$ come from?

It may seem like a dumb question but I'm trying to solve a problem involving coordinate transformations on the Christoffel symbol and to solve it they do the product rule  $$\partial_\alpha g_{\beta ' \gamma '} = \frac{\partial}{\partial x^{\alpha}} \left(\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \right)g_{\beta \gamma} + \frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}.$$

I understand that we need to take the product rule, but I don't understand why the second term is $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}$ and not $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial}{\partial x^{\alpha }} g_{\beta \gamma}$. Where does the $\frac{\partial x^{\nu}}{\partial x^{\alpha }}$ come from?

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Qmechanic ♦
Qmechanic ♦
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It may seem like a dumb question but I'm trying to solve a problem involving coordinate transformations on the Christoffel symbol and to solve it they do the following product rule:

$$\partial_\alpha g_{\beta ' \gamma '} = \frac{\partial}{\partial x^{\alpha}} \left(\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \right)g_{\beta \gamma} + \frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}$$$$\partial_\alpha g_{\beta ' \gamma '} = \frac{\partial}{\partial x^{\alpha}} \left(\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \right)g_{\beta \gamma} + \frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}.$$

I understand that we need to take the product rule, but I don't understand why the second term is $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}$ and not $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial}{\partial x^{\alpha }} g_{\beta \gamma}$. Where does the $\frac{\partial x^{\nu}}{\partial x^{\alpha }}$ come from?

It may seem like a dumb question but I'm trying to solve a problem involving coordinate transformations on the Christoffel symbol and to solve it they do the following product rule:

$$\partial_\alpha g_{\beta ' \gamma '} = \frac{\partial}{\partial x^{\alpha}} \left(\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \right)g_{\beta \gamma} + \frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}$$

I understand that we need to take the product rule, but I don't understand why the second term is $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}$ and not $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial}{\partial x^{\alpha }} g_{\beta \gamma}$. Where does the $\frac{\partial x^{\nu}}{\partial x^{\alpha }}$ come from?

It may seem like a dumb question but I'm trying to solve a problem involving coordinate transformations on the Christoffel symbol and to solve it they do the following product rule:

$$\partial_\alpha g_{\beta ' \gamma '} = \frac{\partial}{\partial x^{\alpha}} \left(\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \right)g_{\beta \gamma} + \frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}.$$

I understand that we need to take the product rule, but I don't understand why the second term is $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}$ and not $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial}{\partial x^{\alpha }} g_{\beta \gamma}$. Where does the $\frac{\partial x^{\nu}}{\partial x^{\alpha }}$ come from?

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Samuel Jaramillo
Samuel Jaramillo
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Christoffel symbol and partial derivative of the metric

It may seem like a dumb question but I'm trying to solve a problem involving coordinate transformations on the Christoffel symbol and to solve it they do the following product rule:

$$\partial_\alpha g_{\beta ' \gamma '} = \frac{\partial}{\partial x^{\alpha}} \left(\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \right)g_{\beta \gamma} + \frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}$$

I understand that we need to take the product rule, but I don't understand why the second term is $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial x^{\nu}}{\partial x^{\alpha }} \partial_{\nu} g_{\beta \gamma}$ and not $\frac{\partial x^{\gamma}}{\partial x^{\gamma '}} \frac{\partial x^{\beta}}{\partial x^{\beta '}} \frac{\partial}{\partial x^{\alpha }} g_{\beta \gamma}$. Where does the $\frac{\partial x^{\nu}}{\partial x^{\alpha }}$ come from?