Differential calculus over commutative algebras

In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are: The whole topological information of a smooth manifold M {\displaystyle M} is encoded in the algebraic properties of its R {\displaystyle \mathbb {R} } -algebra of smooth functions A = C ∞ ( M ) , {\displaystyle A=C^{\infty }(M),} as in the Banach–Stone theorem.

Source: Wikipedia — Differential calculus over commutative algebras (CC BY-SA 4.0)

Differential calculus over commutative algebras

In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are: The whole topological information of a smooth manifold M {\displaystyle M} is encoded in the algebraic properties of its R {\displaystyle \mathbb {R} } -algebra of smooth functions A = C ∞ ( M ) , {\displaystyle A=C^{\infty }(M),} as in the Banach–Stone theorem.

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Source: Wikipedia "Differential calculus over commutative algebras" · CC BY-SA 4.0

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