Tensor Variation with Respect to Metric in First Order Formalism

[Chris Harrison]

Homework Statement

I'm just wondering if I'm doing this calculation correct?
eta and f are both tensors

Homework Equations

The Attempt at a Solution

$$\frac{\delta \left ( \gamma_{3}f{_{\lambda}}^{k}f{_{k}}^{\sigma}f{_{\sigma}}^{\lambda} \right )}{\delta f^{\mu\nu}}=\frac{\delta\left (\gamma_{3} f^{\epsilon k}\eta_{\lambda\epsilon}f^{\rho\sigma}\eta_{k\rho}f^{\omega\lambda}\eta_{\sigma\omega} \right ) }{\delta f^{\mu\nu}}\\<br /> =\gamma_{3}\left ( \delta_{\mu}^{\epsilon}\delta_{\nu}^{k}f^{\rho\sigma}f^{\omega\lambda}+\delta_{\mu}^{\rho}\delta_{\nu}^{\sigma}f^{\epsilon k}f^{\omega\lambda}+\delta_{\mu}^{\omega}\delta_{\nu}^{\lambda}f^{\epsilon k}f^{\rho\sigma} \right )\times\left ( \eta_{\lambda\epsilon}\eta_{k\rho}\eta_{\sigma\omega} \right )\\<br /> =\gamma_{3}\left ( f{_{\nu}}^{\sigma}f{_{\sigma}}^{\lambda}\eta_{\lambda\mu}+f{_{\nu}}^{\lambda}f{_{\lambda}}^{k}\eta_{k\mu}+f{_{\nu}}^{k}f{_{k}}^{\sigma}\eta_{\sigma\mu} \right )\\<br /> =3\gamma_{3} f{_{\nu}}^{\sigma}f{_{\sigma}}^{\lambda}\eta_{\lambda\mu}$$

 

[fzero]

It looks ok if this is a 1st order formalism where the metric is being expanded around the flat metric: $g_{\mu\nu}=\eta_{\mu\nu} + f_{\mu\nu}$. If it is something else, it may or may not be correct.