2nd Edition
Calculus in Vector Spaces, Revised Expanded
Preface to the Second Edition
Preface to the Third Edition
Some Preliminaries
The Rudiments of Set Theory
Some Logic
Mathematical Induction
Inequalities and Absolute Value
Equivalence Relations
Vector Spaces
The Cartesian Plane
The Definition of a Vector Space
Some Elementary Properties of Vectors Spaces
Subspaces
Linear Transformations
Linear Transformations on Education Spaces
The Derivative
Normed Vector Spaces
Open and Closed Sets
Continuous Functions Between Normed Vector Spaces
Elementary Properties of Continuous Functions
The Derivative
Elementary Properties of the Derivative
Partial Derivatives and the Jacobian Matrix
The Structure of Vector Spaces
Spans and Linear Independence
Bases
Bases and Linear Transformations
The Dimension of a Vector Space
Inner Product Spaces
The Norm on an Inner Product Space
Orthonormal Bases
The Cross Product in R3
Compact and Connected Sets
Convergent Sequences
Compact Sets
Upper and Lower Bounds
Continuous Functions on Compact Sets
A Characterization of Compact Sets
Uniform Continuity
Connected Sets
The Chain Rule, Higher Derivatives, and Taylor’s Theorem
The Chain Rule
Proof of the Chain Rule
Higher Derivatives
Taylor’s Theorem for Functions of One Variable
Taylor’s Theorem for Functions of Two Variables
Taylor’s Theorem for Functions of n Variables
A Sufficient Condition for Differentiability
The Equality of Mixed Partial Derivatives
Linear Transformations and Matrices
The Matrix of a Linear Transformation
Isomorphisms and Invertible Matrices
Change of Basis
The Rank of a Matrix
The Trace and Adjoint of a Linear Transformation
Row and Column Operations
Gaussian Elimination
Maxima and Minima
Maxima and Minima at Interior Points
Quadratic Forms
Criteria for Local Maxima and Minima
Constrained Maxima and Minima: I
The Method of Lagrange Multipliers
Constrained Maxima and Minima: II
The Proof of Proposition 2.3
The Inverse and Implicit Function Theorems
The Inverse Function Theorem
The Proof of Theorem 1.3
The Proof of the General Inverse Function Theorem
The Implicit Function Theorem: I
The Implicit Function Theorem: II
The Spectral Theorem
Complex Numbers
Complex Vector Spaces
Eigenvectors and Eigenvalues
The Spectral Theorem
Determinants
Properties of the Determinant
More on Determinants
Quadratic Forms
Integration
Integration of Functions of One Variable
Properties of the Integral
The Integral of a Function of Two Variables
The Integral of a Function of n Variables
Properties of the Integral
Integrable Functions
The Proof of Theorem 6.2
Iterated Integrals and the Fubini Theorem
The Fubini Theorem
Integrals Over Nonrectangular Regions
More Examples
The Proof of Fubini’s Theorem
Differentiating Under the Integral Sign
The Change of Variable Formula
The Proof of Theorem 6.2
Line Integrals
Curves
Line Integrals of Functions
Line Integrals of Vector Fields
Conservative Vector Fields
Green’s Theorem
The Proof of Green’s Theorem
Surface Integrals
Surfaces
Surface Area
Surface Integrals
Stokes’ Theorem
Differential Forms
The Algebra of Differential Forms
Basic Properties of the Sum and Product of Forms
The Exterior Differential
Basic Properties of the Exterior Differential
The Action of Differentiable Functions on Forms
Further Properties of the Induced Mapping
Integration of Differential Forms
Integration of Forms
The General Stokes’ Theorem
Green’s Theorem and Stokes’ Theorem
The Gauss Theorem and Incompressible Fluids
Proof of the General Stokes’ Theorem
Appendix 1. The Existence of Determinants
Appendix 2. Jordan Canonical Form
1. Generalized Eigenvalues
2. The Jordan Canonical Form
3. Polynomials and Linear Transformations
4. The Proof of Theorem 3.5
5. The Proof of Theorem 2.2
Solutions of Selected Exercises
Index
Biography
Corwin, Lawrence
