Chapter 11 provides a more mathematical derivation of the field equations from a variational principle as was first suggested by Hilbert. Many tensor identities are best derived using the technique of geodesic coordinates. As well as considering the action to be dependent on the metric, this chapter introduces the Palatini approach which depends separately on a connection and a metric. The chapter ends by briefly introducing a matter term into the Lagrangian in order to obtain the Einstein Lagrangian and then the full Einstein field equations. This sets the scene for the next chapter, which considers the energy–momentum tensor.