Several Variable calculus

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