2nd Edition

# Calculus in Vector Spaces, Revised Expanded

**Also available as eBook on:**

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

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 R^{3}

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