Inmathematics thedifferential calculus over commutative algebras is a part ofcommutative algebra based on the observation that most concepts known from classical differentialcalculus can be formulated in purely algebraic terms. Instances of this are:

1. The whole topological information of asmooth manifold${\displaystyle M}$ is encoded in the algebraic properties of its${\displaystyle \mathbb{R}}$-algebra of smooth functions${\displaystyle A=C^{\mathrm{\infty }}(M),}$ as in theBanach–Stone theorem.
2. Vector bundles over${\displaystyle M}$ correspond to projective finitely generatedmodules over${\displaystyle A,}$ via thefunctor${\displaystyle \mathrm{\Gamma }}$ which associates to a vector bundle its module of sections.
3. Vector fields on${\displaystyle M}$ are naturally identified withderivations of the algebra${\displaystyle A}$.
4. More generally, alinear differential operator of order k, sending sections of a vector bundle${\displaystyle E\to M}$ to sections of another bundle${\displaystyle F\to M}$ is seen to be an${\displaystyle \mathbb{R}}$-linear map${\displaystyle \mathrm{\Delta }:\mathrm{\Gamma }(E)\to \mathrm{\Gamma }(F)}$ between the associated modules, such that for any${\displaystyle k+1}$ elements${\displaystyle f_0,\dots ,f_k\in A}$:

$${\displaystyle \left[f_k\left[f_{k-1}\left[\cdots \left[f_0,\mathrm{\Delta }\right]\cdots \right]\right]\right]=0}$$where the bracket${\displaystyle [f,\mathrm{\Delta }]:\mathrm{\Gamma }(E)\to \mathrm{\Gamma }(F)}$ is defined as the commutator$${\displaystyle [f,\mathrm{\Delta }](s)=\mathrm{\Delta }(f\cdot s)-f\cdot \mathrm{\Delta }(s).}$$

Denoting the set of${\displaystyle k}$^th order linear differential operators from an${\displaystyle A}$-module${\displaystyle P}$ to an${\displaystyle A}$-module${\displaystyle Q}$ with${\displaystyle {\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_k(P,Q)}$ we obtain a bi-functor with values in thecategory of${\displaystyle A}$-modules. Other natural concepts of calculus such asjet spaces,differential forms are then obtained asrepresenting objects of the functors${\displaystyle {\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_k}$ and related functors.

Seen from this point of view calculus may in fact be understood as the theory of these functors and their representing objects.

Replacing the real numbers${\displaystyle \mathbb{R}}$ with anycommutative ring, and the algebra${\displaystyle C^{\mathrm{\infty }}(M)}$ with any commutative algebra the above said remains meaningful, hence differential calculus can be developed for arbitrary commutative algebras. Many of these concepts are widely used inalgebraic geometry,differential geometry andsecondary calculus. Moreover, the theory generalizes naturally to the setting ofgraded commutative algebra, allowing for a natural foundation of calculus onsupermanifolds,graded manifolds and associated concepts like theBerezin integral.

• Secondary calculus and cohomological physics – Modern discipline
• Differential algebra – Algebraic study of differential equations
• Spectrum of a ring – Set of a ring's prime ideals

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• Commutative algebra
• Differential calculus