Elements of Differential Calculus over Commutative Algebras

The following sections are included:

• Algebraic Tools

  • General Conventions

  • Differential Operators

  • Derivations

  • Additive Functions on Tensor Products

  • Some Basic Facts

  • Equivalence of Categories

• Smooth Manifolds

  • Dual Space

  • Geometric Algebras

  • Natural Topology

  • Dual Map

  • Restriction Algebra

  • Smooth Algebras

  • C^∞ –closed Algebras

  • Smooth Manifolds

  • Smooth Maps

  • Associated Homomorphism

  • Smoothness Condition

  • Classical Definition of Manifolds

  • Associated Atlas

  • The Equivalence ∊: 𝔖𝔐_a → 𝔖𝔐_c

  • Submanifolds

  • Restrictions

  • Smooth Envelope

  • Smooth Tensor Product

  • Cartesian Product

  • Embeddings of Factors of a Product

  • Tangent Vectors

  • Differential of a Smooth Map

  • Local Description of Submanifolds

  • Closed Embeddings

• Vector Bundles

  • Pseudobundles

  • Geometric Modules

  • Vector Bundles

  • Morphisms of Vector Bundles over the Same Base

  • Induced Bundle

  • Pull-back by a Composition of Maps

  • Morphisms of Vector Bundles

  • Composition of Morphisms

  • Associated Homomorphism with a Regular Morphism of Vector Bundles

  • Universal Property of the Induced Bundle

  • Standard Trivial Bundles

  • Identification Isomorphisms

  • Uniform Morphisms

  • Geometric Definition of Vector Bundles

  • Vector Bundle Morphisms from a Geometric Viewpoint

  • Module of Smooth Sections of a Geometrically Defined Vector Bundle

  • The Total Space as a Smooth Manifold

  • Geometric Criterion for Vector Bundle Morphisms

  • Equivalence Between 𝔙𝔅_g and 𝔙𝔅

• Vector Fields

  • Smooth Vector Fields

  • Image of a Vector Field

  • Compatibility

  • Commutator of Vector Fields

  • Restriction on Open Submanifolds

  • Extension from Closed Submanifolds

  • Tangent Bundle

  • Vector Fields Along Maps

  • Local Vector Fields

  • Splitting of Vector Fields on a Product Manifold

  • Trajectories

• Differential Forms

  • Differential Forms with Values in a Module

  • Cotangent Bundle

  • The Ordinary Differential

  • Universal Property of First Order Differential Forms

  • Action of Smooth Maps on 1-Forms

  • Geometric Description of Λ¹ (f^*)

  • Exterior Differential

  • de Rham Complex

  • Wedge Product

  • Leibnitz Rule for the Exterior Differential

  • Action of Smooth Maps on de Rham Complexes

  • Geometric Description of Λ^• (f^*)

  • Insertion Operator

  • Leibnitz Rule for Insertion Operators

  • Differential Forms Along Maps

  • Wedge Product of Differential Forms Along Maps

  • Insertion Along Maps

  • Leibnitz Rule for Insertions Along Maps

  • Local Differential Forms

  • Splitting of Differential Forms on a Product Manifold

  • Splitting of Derivations

  • Splitting of Tangent Vectors

• Lie Derivative

  • Flow Generated by a Vector Field

  • Flows on a Manifold with Boundary

  • Relative Intervals

  • Smooth Dependence on the Initial Data

  • One-parameter Group Generated by a Vector Field

  • Smooth Families of Smooth Maps

  • Smooth Families of Vector Fields

  • Time-dependent Vector Field Associated with a One-parameter Family of Diffeomorphisms

  • Smooth Families of Differential Forms

  • Derivative of a Time-dependent Differential Form

  • Lie Derivative of Differential Forms

  • Cartan Formula

  • Leibnitz Rule for Lie Derivatives