be obtained by standard textbook calculations such as the one given below, which is strongly based on the one given in Carroll (2004). Variation of the Riemann tensor, the Ricci tensor, and the Ricci scalar To calculate the variation of the Ricci scalar we calculate first the variation of the Riemann curvature tensor, and then the variation of the Ricci tensor. So, the Riemann curvature tensor is defined as . Since the Riemann curvature depends only on the Levi-Civita connection , the variation of the Riemann tensor can be calculated as . Now, since is the difference of two connections, it is a tensor and we can thus calculate its covariant derivative, . We can now observe that the expression for the variation of Riemann curvature tensor above is equal to the difference of two such terms, . We may now obtain the variation of the Ricci curvature tensor simply by contracting two indices of the variation of the Riemann tensor, and get the Palatini identity: . The Ricci scalar is defined as . Therefore, its variation with respect to the inverse metric is given by In the second line we used the metric compatibility of the covariant derivative, , and the previously obtained result for the variation of the Ricci curvature (in the second term, renaming the dummy indices and to and respectively). The last term, , i.e. with , multiplied by , becomes a total derivative, since for any vector and any tensor density we have: or and thus by Stokes' theorem only yields a boundary term when integrated. The boundary term is in general non-zero, because the integrand depends not only on but also on its partial derivatives ; see the article Gibbons–Hawking–York boundary term for details. However when the variation of the metric vanishes in a neighbourhood of the boundary or when there is no boundary, this term does not contribute to the variation of the action. And we thus obtain at events not in the closure of the boundary. Variation of the determinant Jacobi's formula, the rule for differentiating a determinant, gives: , or one could transform to a coordinate system where is diagonal and then apply the product rule to differentiate the product of factors on the main diagonal. Using this we get In the last equality we used the fact that which follows from the rule for differentiating the inverse of a matrix