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as the map from points x ∈ R to unit m -vectors corresponding to the oriented tangent plane to R at x . By this, we mean that S ( x ) is the wedge product of m orthonormal tangent vectors to R at x . Then for any differential form φ ∈ D $^{m}$( R $^{n}$), define an m - current T by 

<img src='content_image/312830.jpg'>

We allow T to carry integer multiplicities by introducing a function η : R → Z with $^{∫}$$_{R}$η ( x ) d H m < ∞ to obtain 

<img src='content_image/312831.jpg'>

The currents which can be constructed via this procedure are called rectifiable currents and their existence justifies the statement that currents generalize oriented submanifolds. 

Definition 1.3.9 (Rectifiable currents) . A rectifiable current is a current with compact support associated with a rectifiable set with integer multiplicities and finite total measure (counting multiplicities). 

Definition 1.3.10 (Normal currents) . An m -current T is normal if and only if M( T ) + M( ∂T ) < ∞ and the support of T is compact. 

Note that nothing prevents normal currents from being “smeared” out in space. Morgan [32, p. 48] gives an example of a normal 1-current S$_{2}$ which covers the unit square in R 2 but with the concentration of mass of the 2-dimensional Hausdorff measure so it has finite mass and boundary. 

Definition 1.3.11 (Integral currents) . A current T is an integral current if T and ∂T are rectifiable. 

As an aid to understanding the various classes of currents, note that all integral currents are both normal and rectifiable (in fact, this can be taken as the definition of 

