Differential of The First Kind

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High Quality Content by WIKIPEDIA articles! In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic geometry), for everywhere-regular differential 1-forms. Given a complex manifold M, a differential of the first kind is therefore the same thing as a 1-form that is everywhere holomorphic; on an algebraic variety V that is non-singular it would be a global section of the coherent sheaf 1 of Kähler differentials. In either case the definition has its origins in the theory of abelian integrals.