“A k‑dimensional current in an open subset U of ℝⁿ is a continuous linear functional on the space of smooth k‑forms with compact support in U. The boundary of a k‑current is defined by duality with the exterior derivative. The mass of a current is the supremum of its values on forms of pointwise norm ≤ 1.”
Generalized surfaces defined as linear functionals on differential forms. They combine the topological structure of chains with the analytic power of distributions. federer geometric measure theory pdf
Federer introduced the concept of , which are a generalization of manifolds (like chains in homology theory) that allow for weak limits of surfaces. The area formula and coarea formula are foundational for integrating functions over these generalized, rectifiable sets. Why "Federer GMT" Remains Essential “A k‑dimensional current in an open subset U