Several Variable calculus

Description

Agenda Things we need to decide What do we do in this course? Normed Linear Spaces Metric Spaces Examples from geometry Euclidean Metric Spherical metric Normed Linear Spaces Continuity Topology of a metric space Compact subsets of a finite dimensional vector space Continuous Linear Maps Bounded Linear Maps Equivalence of Norms on Finite Dimensional Vector Spaces Completeness, Contractions, Consequences Lipschitz Maps The Classification Question Series in Banach Spaces Contraction Principle Applications of Contraction Theorem Exponential, Logarithm and other Functions Bounded Linear maps as a normed linear space The General Linear Group of a Banach Space The exponential of a linear map Logarithm and other functions Differentiation The Gâteaux Derivative The Fréchet Derivative Gâteaux Differential as Velocity Elementary Properties of Gâteaux/ Fréchet Derivative Application: Conservation of Angular Momentum Comparison of Gâteaux and Fréchet derivative The Jacobian (Operator) Matrix Differentiating Maps to a Direct Sum Differentiating Maps defined on an Open Subset of a Direct Sum Combining the Two: Maps from an Open Subset of a Direct Sum to a Direct Sum The Integral The Cauchy Integral Characterization of Regulated Functions Characterization of the Cauchy Integral The Riemann Integral Properties of the Integral Taylor's Formula Fundamental Theorem of Calculus Symmetries of Higher Derivatives Taylor's Formula Applications I: Linear Lie Groups Properties of the Exponential Map Lie Algebra of a Linear Lie Group Lie Algebras of Some Linear Lie Groups Lie Group Lie Algebra Correspondence BCH Formula Integrating Lie Algebra Homomorphisms Simply Connected Linear Lie Groups Isomorphism Theorems Inverse Function Theorem Corollaries of Inverse Function Theorem Implicit Function Theorem Ordinary Differential Equations Reduction to time/parameter independent case The time dependent case The time and parameter dependent case Existence and Uniqueness of Flows for Lipschitz Vector Fields Differentiability of Dependence on Initial Conditions The Flow Equation Vector Fields as Derivations Applications: Optimization Unconstrained Otimizaton Critical Points and Gradient Flow Constrained Optimization: Necessary Condition Constrained Optimization: Sufficient Condition Geometric Description of Lagrange Rule Change of Variables in Multiple Integrals Multilinear Algebra tensor Products G-Spaces Alternating/Exterior/Wedge Product of Vector Spaces The Wedge/Antisymmetric product Map A Second Approach to Wedge Product Map Pairing Between Alternating Forms and Antisymmetric Product Appendix Axiom of Choice and All That! Appendix Hahn-Banach Theorems Appendix Homotopy Fundamental Group Covering Space Miscellaneous Exercises