Differential calculus | Commutative algebra
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: 1. * The whole topological information of a smooth manifold is encoded in the algebraic properties of its -algebra of smooth functions as in the Banach–Stone theorem. 2. * Vector bundles over correspond to projective finitely generated modules over via the functor which associates to a vector bundle its module of sections. 3. * Vector fields on are naturally identified with derivations of the algebra . 4. * More generally, a linear differential operator of order k, sending sections of a vector bundle to sections of another bundle is seen to be an -linear map between the associated modules, such that for any elements : where the bracket is defined as the commutator Denoting the set of th order linear differential operators from an -module to an -module with we obtain a bi-functor with values in the category of -modules. Other natural concepts of calculus such as jet spaces, differential forms are then obtained as representing objects of the functors 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 with any commutative ring, and the algebra 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 in algebraic geometry, differential geometry and secondary calculus. Moreover, the theory generalizes naturally to the setting of graded commutative algebra, allowing for a natural foundation of calculus on supermanifolds, graded manifolds and associated concepts like the Berezin integral. (Wikipedia).
Calculus 3 Lecture 13.4: Finding Differentials of Multivariable Functions
Calculus 3 Lecture 13.4: Finding Differentials of Multivariable Functions: A review of Differentials from Calculus 1 and an extrapolation towards Differentials with more than 1 Independent Variable. Focus will be on the derivation of the idea of Differentials and the application of Diff
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Schemes 46: Differential operators
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. In this lecture we define differential operators on rings, and calculate the universal (normalized) differential operator of order n. As a special case we fin
From playlist Algebraic geometry II: Schemes
Differentiating a Continued Fraction
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Branimir Cacic, Classical gauge theory on quantum principalbundles
Noncommutative Geometry Seminar (Europe), 20 October 2021
From playlist Global Noncommutative Geometry Seminar (Europe)
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The video introduces higher order linear differential equations and related theorems on superposition, existence and uniqueness, and linear independence. https://mathispower4u.com
From playlist Differential Equations: Complete Set of Course Videos
Kevin Buzzard (lecture 18/20) Automorphic Forms And The Langlands Program [2017]
Full course playlist: https://www.youtube.com/playlist?list=PLhsb6tmzSpiysoRR0bZozub-MM0k3mdFR http://wwwf.imperial.ac.uk/~buzzard/MSRI/ Summer Graduate School Automorphic Forms and the Langlands Program July 24, 2017 - August 04, 2017 Kevin Buzzard (Imperial College, London) https://w
From playlist MSRI Summer School: Automorphic Forms And The Langlands Program, by Kevin Buzzard [2017]
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From playlist Calculus 2 (Full Length Videos)
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From playlist Spring 2022 Online Kolchin seminar in Differential Algebra
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Erik van Erp: Lie groupoids in index theory 4
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Shahn Majid: Quantum geodesic flows and curvature
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From playlist Global Noncommutative Geometry Seminar (Europe)
Masoud Khalkhali: Curvature of the determinant line bundle for noncommutative tori
I shall first survey recent progress in understanding differential and conformal geometry of curved noncommutative tori. This is based on work of Connes-Tretkoff, Connes-Moscovici, and Fathizadeh and myself. Among other results I shall recall the computation of spectral invariants, includi
From playlist HIM Lectures: Trimester Program "Non-commutative Geometry and its Applications"
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