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In this paper we try to construct noncommutative Yang-Mills theory forgeneric Poisson manifolds. It turns out that the noncommutative differentialcalculus defined in an old work is exactly what we need. Using this calculus,we generalize results about the Seiberg-Witten map, the Dirac-Born-Infeldaction, the matrix model and the open string quantization for constant B fieldto non-constant background with H=0.Comment: 21 pages, Latex file, references added, minor modificatio

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

Noncommutative differential calculus for a D-brane in a nonconstant<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi></mml:math>field background with<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>H</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mn /></mml:math>

Noncommutative differential calculus for a D-brane in a nonconstantBfield background withH=0