The Noether theorems were derived by Noether for n-dimensional Euclidean spaces, but they have been used by many writers in relativistic theories where the geometry is not Euclidean. We give a derivation of the Noether theorems, assuming only a Riemannian space and following the method used by Noether as closely as possible. This requires new definitions of total variations for fields and integrals since a covariant total variation for tensor fields is required. The results have been applied to electromagnetic fields.