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THE GEOMETRY OF MATHEMATICAL METHODS
Corinne A. Manogue, Tevian Dray
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Contents
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Front Matter
Colophon
1
Coordinates and Vectors
Polar Coordinates
Curvilinear Coordinates
Change of Coordinates
Vectors
Bases
Unit Vectors
The Dot Product
Visualizing the Dot Product
The Law of Cosines
Addition Formulas
Orthonormal Basis Vectors
Polar basis vectors
Orthonormality of Basis Vectors
The Position Vector as a Vector Field
The Position Vector in Curvilinear Coordinates
The Distance Formula
Scalar Fields
Vector Fields
The Cross Product
Lines and Planes
Linearity of the Dot and Cross Products
2
Complex Numbers
The Complex Plane
Complex Conjugate and Norm
Algebra with Complex Numbers: Rectangular Form
Division: Rectangular Form
Euler's Formula
Exponential Form
Sums of Harmonic Functions
Roots of Complex Numbers
Logarithms of Complex Numbers
3
Operations with Matrices
Matrix Addition
Scalar Multiplication
Matrix Multiplication
Transpose
Hermitian Adjoint
Dot Products
Inner Products for Complex Vectors
Trace
Determinants
Inverses
4
Eigenvectors and Eigenvalues
What are Eigenvectors?
Finding Eigenvalues
Finding Eigenvectors
Normalization of Eigenvectors
Diagonal Matrices
Degeneracy
Using Eigenvectors as a Natural Basis
5
Special Matrices
Hermitian Matrices
Properties of Hermitian Matrices
Commuting Matrices
Properties of Unitary Matrices
Unitary Matrices
Change of Basis
Symmetry Operations
Matrix Examples
Matrix Decompositions
Matrix Exponentials
Evolution Equation
6
Differentials
Review of Single Variable Differentiation
Thick Derivatives
Leibniz vs. Newton
Differentials
Rules for Differentials
Properties of Differentials
Substitution
Differentials: Summary
The Multivariable Differential
Chain Rule
Chain Rule via Tree Diagrams
Applications of Chain Rule
Interpreting Differentials
Things not to do with Differentials
7
Power Series
Definition of Power Series
Calculating Power Series Coefficients
Visualization of Power Series Approximations
Discussion of Approximations using Power Series
Using Technology to Explore Power Series Approximations
Guessing Power Series from Graphs
Common Power Series
Dimensions in Power Series
Convergence of Power Series
Theorems about Power Series
8
Differential Equations
Definitions and Theorems
First Order ODEs: Notation and Theorems
Separable ODEs
Exact ODEs
The Word “Linear”: Definitions and Theorems
Theorems about Linear ODEs
Constant Coefficients, Homogeneous
Linear Independence
Constant Coefficients, Inhomogeneous
Power Series Solutions: Theorem
Power Series Solutions: Method/Example
9
Vector Spaces
Definition of a Vector Space
Definition and Properties of an Inner Product
Visualizing the Dot Product in Higher Dimensions
Visualizing the Product of Two Functions
Inner Products of Functions
Linear Operators
10
Fourier Series
Fourier Series Motivation
Fourier Basis Functions
Inner Products of Harmonic Functions
Completeness
Fourier Coefficients
Fourier Series Example
Fourier Series: Worked Example
Fourier Series: Exploration
Fourier Series: Small Group Activity
The Gibbs Phenomenon
Symmetries
Fourier Coefficients in Space and Time
11
Delta Functions
Step Functions
The Dirac Delta Function
Properties of the Dirac Delta Function
Representations of the Dirac Delta Function
The Dirac Delta Function in Three Dimensions
The Exponential Representation of the Dirac Delta Function
12
Fourier Transforms
Gaussians
Normalization of the Gaussian
Definition of the Fourier Transform
Fourier Transform of the Delta Function
Properties of the Fourier Transform
Examples of Fourier Transforms
Using Technology to Calculate and Graph Fourier Transforms
Fourier Uncertainties
Wave Packets
13
Partial Differential Equations
Important PDEs in Physics
Classification of PDEs
PDE Theorems
Separation of Variables
Sturm–Liouville Theory
14
Classical Mechanics of Orbits
Graphs in Polar Coordinates
Effective Potential
Effective Potential and Orbits
15
Quantum Mechanics
States on a Ring
Legendre Expansions
Back Matter
A
Appendix
Completing the Square
Bibliography
Authored in PreTeXt
Section
5.7
Symmetry Operations