2nd Edition

Calculus in Vector Spaces, Revised Expanded

By Lawrence Corwin, Robert Szczarba Copyright 1995

    Calculus in Vector Spaces addresses linear algebra from the basics to the spectral theorem and examines a range of topics in multivariable calculus. This second edition introduces, among other topics, the derivative as a linear transformation, presents linear algebra in a concrete context based on complementary ideas in calculus, and explains differential forms on Euclidean space, allowing for Green's theorem, Gauss's theorem, and Stokes's theorem to be understood in a natural setting. Mathematical analysts, algebraists, engineers, physicists, and students taking advanced calculus and linear algebra courses should find this book useful.

    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


    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 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


    Properties of the Determinant

    More on Determinants

    Quadratic Forms


    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


    Line Integrals of Functions

    Line Integrals of Vector Fields

    Conservative Vector Fields

    Green’s Theorem

    The Proof of Green’s Theorem

    Surface Integrals


    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



    Corwin, Lawrence