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I am having the following action with an induced metric and a scalar field: $$S = \int dr d^dx \, \left( \partial_r \phi^2 + \frac{1}{2}

\gamma^{ij} \partial_i \phi \partial_j \phi + V(\phi) \right)$$

Now, I am looking at a paper (which has a slightly more complicated action, and I want to vary this action. Can someone explain in a nice way how to do this since I have some problems? I need a variation w.r.t. the field and the radial coordinare r.

Also can someone summarize the general way to vary an action with various fields and derivatives of fields?

Thank you very much!

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# Variation of an action with a background metric

Can you offer guidance or do you also need help?

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