Basic Analysis II: A Modern Calculus in Many Variables

Cover
Half Title
Title Page
Copyright Page
Dedication Page
Table of Contents
I Introduction
1 Beginning Remarks
1.1 Table of Contents
1.2 Acknowledgments
II Linear Mappings
2 Preliminaries
2.1 The Basics
2.2 Some Topology in ℜ2
2.2.1 Homework
2.3 Bolzano - Weierstrass in ℜ2
2.3.1 Homework
3 Vector Spaces
3.1 Vector Spaces over a Field
3.1.1 The Span of a Set of Vectors
3.2 Inner Products
3.2.1 Homework
3.3 Examples
3.3.1 Two Dimensional Vectors in the Plane
3.3.2 The Connection between Two Orthonormal Bases
3.3.3 The Invariance of the Inner Product
3.3.4 Two Dimensional Vectors as Functions
3.3.5 Three Dimensional Vectors in Space
3.3.6 The Solution Space of Higher Dimensional ODE Systems
3.4 Best Approximation in a Vector Space with Inner Product
3.4.1 Homework
4 Linear Transformations
4.1 Organizing Point Cloud Data
4.1.1 Homework
4.2 Linear Transformations
4.2.1 Homework
4.3 Sequence Spaces Revisited
4.3.1 Homework
4.4 Linear Transformations between Normed Linear Spaces
4.4.1 Basic Properties
4.4.2 Mappings between Finite Dimensional Vector Spaces
4.5 Magnitudes of Linear Transformations
5 Symmetric Matrices
5.1 The General Two by Two Symmetric Matrix
5.1.1 Examples
5.1.2 A Canonical Form
5.1.3 Two Dimensional Rotations
5.1.4 Homework
5.2 Rotating Surfaces
5.2.1 Homework
5.3 A Complex ODE System Example
5.3.1 The General Real and Complex Solution
5.3.2 Rewriting the Real Solution
5.3.3 Signed Definite Matrices
5.3.4 Summarizing
5.4 Symmetric Systems of ODEs
5.4.1 Writing the Solution Another Way
5.4.2 Homework
6 Continuity and Topology
6.1 Topology in n Dimensions
6.1.1 Homework
6.2 Cauchy Sequences
6.3 Compactness
6.3.1 Homework
6.4 Functions of Many Variables
6.4.1 Limits and Continuity for Functions of Many Variables
7 Abstract Symmetric Matrices
7.1 Input-Output Ratios for Matrices
7.1.1 Homework
7.2 The Norm of a Symmetric Matrix
7.2.1 Constructing Eigenvalues
7.3 What Does This Mean?
7.4 Signed Definite Matrices Again
7.4.1 Homework
8 Rotations and Orbital Mechanics
8.1 Introduction
8.1.1 Homework
8.2 Orbital Planes
8.2.1 Orbital Constants
8.2.2 The Orbital Motion
8.2.3 The Constant B Vector
8.2.4 The Orbital Conic
8.3 Three Dimensional Rotations
8.3.1 Homework
8.3.2 Drawing Rotations
8.3.3 Rotated Ellipses
8.4 Drawing the Orbital Plane
8.4.1 The Perifocal Coordinate System
8.4.2 Orbital Elements
8.5 Drawing Orbital Planes Given Radius and Velocity Vectors
8.5.1 Homework
9 Determinants and Matrix Manipulations
9.1 Determinants
9.1.1 Consequences One
9.1.2 Homework
9.1.3 Consequences Two
9.1.4 Homework
9.2 Matrix Manipulation
9.2.1 Elementary Row Operations and Determinants
9.2.2 Code Implementations
9.2.3 Matrix Inverse Calculations
9.3 Back to Definite Matrices
III Calculus of Many Variables
10 Differentiability
10.1 Partial Derivatives
10.1.1 Homework
10.2 Tangent Planes
10.3 Derivatives for Scalar Functions of n Variables
10.3.1 The Chain Rule for Scalar Functions of n Variables
10.4 Partials and Differentiability
10.4.1 Partials Can Exist but Not be Continuous
10.4.2 Higher Order Partials
10.4.3 When Do Mixed Partials Match?
10.5 Derivatives for Vector Functions of n Variables
10.5.1 The Chain Rule for Vector-Valued Functions
10.6 Tangent Plane Error
10.6.1 The Mean Value Theorem
10.6.2 Hessian Approximations
10.7 A Specific Coordinate Transformation
10.7.1 Homework
11 Multivariable Extremal Theory
11.1 Differentiability and Extremals
11.2.1 Positive and Negative Definite Hessians
11.2.2 Expressing Conditions in Terms of Partials
12 The Inverse and Implicit Function Theorems
12.1 Mappings
12.2 Invertibility Results
12.2.1 Homework
12.3 Implicit Function Results
12.3.1 Homework
12.4 Constrained Optimization
12.4.1 What Does the Lagrange Multiplier Mean?
12.4.2 Homework
13 Linear Approximation Applications
13.1 Linear Approximations to Nonlinear ODE
13.1.1 An Insulin Model
13.2 Finite Difference Approximations in PDE
13.2.1 First Order Approximations
13.2.2 Second Order Approximations
13.2.3 Homework
13.3 FD Approximations
13.3.1 Error Analysis
13.3.2 Homework
13.4 FD Diffusion Code
13.4.1 Homework
IV Integration
14 Integration in Multiple Dimensions
14.1 The Darboux Integral
14.1.1 Homework
14.2 The Riemann Integral in n Dimensions
14.2.1 Homework
14.3 Volume Zero and Measure Zero
14.3.1 Measure Zero
14.3.2 Volume Zero
14.4 When is a Function Riemann Integrable?
14.4.1 Homework
14.5 Integration and Sets of Measure Zero
14.5.1 Homework
15 Change of Variables and Fubini’s Theorem
15.1 Linear Maps
15.1.1 Homework
15.2 The Change of Variable Theorem
15.2.1 Homework
15.3 Fubini Type Results
15.3.1 Fubini on a Rectangle
15.3.2 Homework
16 Line Integrals
16.1 Paths
16.1.1 Homework
16.2 Conservative Force Fields
16.2.1 Homework
16.3 Potential Functions
16.3.1 Homework
16.4 Green’s Theorem
16.4.1 Homework
16.5 Green’s Theorem for Images of the Unit Square
16.5.1 Homework
16.6 Motivational Notation
16.6.1 Homework
17 Differential Forms
17.1 One Forms
17.1.1 Smooth Paths
17.1.2 What is a Winding Number?
17.2 Exact and Closed 1-Forms
17.2.1 Smooth Segmented Paths
17.3 Two and Three Forms
V Applications
18 The Exponential Matrix
18.1 The Exponential Matrix
18.2 The Jordan Canonical Form
18.2.1 Homework
18.3 Exponential Matrix Calculations
18.3.1 Jordan Block Matrices
18.3.2 General Matrices
18.3.3 Homework
18.4 Applications to Linear ODE
18.4.1 The Homogeneous Solution
18.4.2 Homework
18.5 The Non-Homogeneous Solution
18.5.1 Homework
18.6 A Diagonalizable Test Problem
18.6.1 Homework
18.7 Simple Jordan Blocks
18.7.1 Homework
19 Nonlinear Parametric Optimization Theory
19.1 The More Precise and Careful Way
19.2 Unconstrained Parametric Optimization
19.3 Constrained Parametric Optimization
19.3.1 Hessian Error Estimates
19.3.2 A First Look at Constraint Satisfaction
19.3.3 Constraint Satisfaction and the Implicit Function Theorem
19.4 Lagrange Multipliers
VI Summing It All Up
20 Summing It All Up
VII References
VIII Detailed Index