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Let S(x) denote the unit m-vector associated with the oriented tangent plane to S at x. Then for any differential m-form ϕ, deﬁne S(x), ϕ d H m . S(ϕ) = S Furthermore, we will allow S to carry a positive integer multiplicity μ(x), with m < ∞, and deﬁne S μ(x) d H S(x), ϕ μ(x) d H m . S(ϕ) = S Finally, we will require that S have compact support. Such currents are called rectiﬁable currents. Normal and Rectiﬁable Currents 41 Deﬁnitions for currents are by duality with forms. The boundary of an m-dimensional current T ∈ Dm is the (m − 1)-dimensional current ∂T ∈ Dm−1 deﬁned by ∂T(ϕ) = T(dϕ).

Deﬁne a function g : Rm → Rn+m x → (f(x), εx). Then Jm (g) ≤ ε(Lip f + ε)m−1 . Now by Case 1, H m (f(A)) ≤ H m (g(A)) = Jm g A ≤ ε(Lip f + ε)m−1 L m (A). Therefore, the right-hand side also must vanish. 7(1) by approximating u by simple functions. The following useful formula relates integrals of a function f over a set A to the areas of the level sets A ∩ f −1 {y} of the function. 11] Consider a Lipschitz functionf : Rm → Rn with m > n. If A is an L m measurable set, then Jn f(x) dL m x = A Rn H m−n (A ∩ f −1 {y}) dL n y.

The oriented m-planes through the origin in Rn are in one-to-one correspondence with the unit, simple m-vectors in m Rn . 3]). Let Rn∗ denote the space of covectors dual to Rn , with dual orthonormal basis ∗ e1 , . . , en∗ . We remark that dxj is a common alternate notation for e∗j . The dual Normal and Rectiﬁable Currents 39 space to m Rn is the space m Rn ≡ m (Rn∗ ) of linear combinations of wedge products of covectors, called m-covectors. The dual basis is {e∗i1 ... im : i1 < · · · < im }. A differential m-form ϕ on Rn is an m-covectorﬁeld; that is, a map ϕ : Rn → m Rn .