Definition 1.3.6 (Currents) . The space of m -currents in$_{R}$ n is denoted by D $_{m}$( R $^{n}$) and is dual to the set of C ∞ differential forms of compact support. The mass of a current T ∈ D $_{m}$( R $^{n}$) is given by Whereas differential forms correspond to integrands, (suitably nice) currents can be intuitively thought of as the linear integration operator itself, representing and gener- alizing the oriented submanifolds over which we can integrate differential forms. When T ∈ D $_{m}$( R $^{n}$) represents an oriented submanifold, the mass is simply its m -dimensional volume, counting multiplicities (this is the intuition take take away from this definition). Definition 1.3.7 (Boundary of a current) . The boundary of an m -current T ∈ D $_{m}$( U ) is defined in terms of the exterior derivative on differential forms. Namely, for m > 0 , we let ∂T ∈ D m − $_{1}$( U ) be the linear operator on ( m − 1) -forms defined by ∂T ( φ ) = T (d φ ) for all φ ∈ D m − $^{1}$( U ) . For m = 0 , we let ∂T = 0 as a 0-current. This definition along with facts about exterior differentiation immediately provides us with some useful properties: Defining ∂T for 0-currents is not universal, but doing so allows us to simplify various statements slightly by omitting special cases (Equation (1.1a), for example). Definition 1.3.8 (Support of a current) . The support of a current T in D m is the complement of the largest open set U such that T ( φ ) = 0 whenever φ ∈ D $^{m}$( U ) . Currents can be created from any oriented m -dimensional rectifiable set R . Define S 7 ∂∂T = 0 , (1.1a) ∂ ( αT$_{1}$ + βT$_{2}$ ) = α∂T$_{1}$ + β∂T$_{2}$. (1.1b) M( T ) = sup { T ( φ ) | φ ∈ D $^{m}$( R ) , φ has compact support, ‖ φ ‖ ≤ 1 } .