- #1

- 129

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[tex]

\delta(\sqrt{\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi})

[/tex]

what is the variation of this quantity under variation of \phi?

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- Thread starter arroy_0205
- Start date

- #1

- 129

- 0

[tex]

\delta(\sqrt{\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi})

[/tex]

what is the variation of this quantity under variation of \phi?

- #2

- 129

- 0

Is the rule here

[tex]

\delta(\sqrt{\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi})=\frac{\partial(\sqrt{\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi})}{\partial \phi}\delta(\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi)

[/tex]

But then the derivative part creates problem. The part under square root depends on derivative of \phi and not on \phi itself, so the result is zero. I am confused.

[tex]

\delta(\sqrt{\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi})=\frac{\partial(\sqrt{\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi})}{\partial \phi}\delta(\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi)

[/tex]

But then the derivative part creates problem. The part under square root depends on derivative of \phi and not on \phi itself, so the result is zero. I am confused.

Last edited:

- #3

- 410

- 0

[tex]\delta F[\phi] = F[\phi + \delta \phi] - F[\phi][/tex].

In any case, in your case the chain rule applies:

[tex]

\delta(\sqrt{\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi})=

\frac{1}{ 2\sqrt{\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi} }

\delta(\eta^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi)

[/tex]

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