In this chapter we introduce the notion of an orienting n-form on an n-dimensional orientable manifold with a metric tensor \(\mathtt {g}\) , the notions of signed measures, homogeneous decomposable p-forms, symmetric non-degenerate bilinear metric tensor fields on p-forms and the linear Hodge map denoted \(\star \) . This map is used to define the covariant operator \(\delta \) on p-forms. For \(\nabla \) a torsion-free metric-compatible connection, it is shown that \(\boldsymbol {\nabla }\star = \star \boldsymbol {\nabla }\) . The Hodge-de Rham operator is defined in Sect. 8.2, integration of differential forms is defined in Sect. 8.3 and Komar 2-forms are defined in Sect. 8.4.