1Introduction to Linear Algebra
163Definition:: Graphical Representation:
21.1 Vectors and Geometric Vectors
16410.2 Line Integrals: Definition of Line Integrals:
3Definition of a Vector:
16510.3 Green’s Theorem
4Representation of Vectors:
166Statement of Green’s Theorem:: Proof and Intuition:
5Operations on Vectors:: Properties of Vector Operations:
16710.4 Divergence Theorem
61.2 Matrix Operations
168Statement of the Divergence Theorem:
7Definition of a Matrix:
169Proof of the Divergence Theorem:: Applications and Examples:
8Representation of Matrices:
17010.5 Stokes’ Theorem
9Types of Matrices:
171Statement of Stokes’ Theorem:
10Operations on Matrices:
172- ∇ × 𝐕 is the curl of 𝐕, which is a vector field: Proof of Stokes’ Theorem:
11Properties of Matrix Operations:
17310.6 Applications in Electromagnetism
12Examples and Applications:
174Fourier Series
131.3 Systems of Linear Equations
17511.1 Periodic Functions
14Definition of a Linear Equation:
176Definition of a Periodic Function:
15Definition of a System of Linear Equations:
177Properties of Periodic Functions:
16Methods for Solving Systems of Linear Equations:
178Examples of Periodic Functions:: Importance of Periodic Functions:
17Properties and Solutions:
17911.2 Fourier Series Expansion
18Applications:
180Fourier Series Representation:
191.4 Gaussian Elimination
181Computation of Fourier Coefficients:: Examples of Fourier Series Expansions:
20Row Operations in Gaussian Elimination:
18211.3 Convergence and Properties
21Steps of Gaussian Elimination:
183Convergence of Fourier Series:
22Properties and Considerations:: Computational Complexity:
184Dirichlet’s Conditions:
231.5 Determinants
185Properties of Fourier Series:
24Definition of a Determinant:
186Applications of Fourier Series:
25Computation of Determinants:
18711.4 Complex Fourier Series
26Properties of Determinants:: Applications of Determinants:
188Complex Fourier Series Representation:
271.6 Inverse of a Matrix
189Computation of Complex Fourier Coefficients:
28Definition of Matrix Inverse:
190Properties of Complex Fourier Series:
29Existence and Uniqueness of Matrix Inverse:
191Advantages of Complex Fourier Series:
30Computation of Matrix Inverse:
192Applications of Complex Fourier Series:
31Properties of Matrix Inverse:
19311.5 Applications in Physics
32Applications of Matrix Inverse:
194Differential Equations
33Vector Spaces
19512.1 First-Order Differential Equations
342.1 Definition and Examples
196General Form:
35Definition of a Vector Space:
197Types of First-Order Differential Equations:
36Examples of Vector Spaces:: Properties and Operations in Vector Spaces:
198Solution Methods:: Applications in Physics:
372.2 Subspaces
19912.2 Second-Order Linear Differential Equations
38Definition of a Subspace:
200General Form:
39Properties of Subspaces:: Examples of Subspaces:
201Characteristic Equation:
402.3 Linear Combinations and Span
202Solution Methods:
41Linear Combinations:
203Non-Homogeneous Equations:
42Properties of Linear Combinations:
204Applications in Physics:
43Span:
20512.3 Systems of Differential Equations
44Properties of Span:
206General Form:
45Applications and Examples:
207Types of Systems:
462.4 Linear Independence
208Solution Methods:
47Definition of Linear Independence:: Testing for Linear Independence:
209Applications in Physics:
482.5 Basis and Dimension
21012.4 Series Solutions
49Basis of a Vector Space:
211Power Series Solutions:
50Properties of a Basis:
212Frobenius Method:: Legendre’s Equation and Legendre Polynomials:
51Dimension of a Vector Space:
21312.5 Laplace Transforms
52Properties of Dimension:
214Definition of the Laplace Transform:
53Finding a Basis and Dimension:
215Properties of the Laplace Transform:
542.6 Change of Basis
216Inverse Laplace Transform:
55Definition of Change of Basis:
217Applications in Differential Equations:
56Properties and Applications of Change of Basis:: Calculating the Change of Basis Matrix:
21812.6 Applications in Physics
57Linear Transformations
219Numerical Methods
583.1 Definition and Properties: Properties of Linear Transformations:
22013.1 Roots of Equations
593.2 Matrix Representation: Algorithms and Pseudocode:
221Bisection Method:
603.3 Kernel and Range
222Newton-Raphson Method:: Secant Method:
61Kernel (Null Space):
22313.2 Interpolation and Approximation
62Properties of the Kernel:
224Polynomial Interpolation:
63Range (Image):
225Lagrange Interpolation Polynomial:
64Properties of the Range:
226Newton’s Divided Difference Interpolation:
65Rank-Nullity Theorem:
227Spline Interpolation:
663.4 Isomorphisms
228Cubic Spline Interpolation:
67Properties of Isomorphisms:: Examples:
229Least Squares Approximation:
683.5 Composition of Linear Transformations
230Linear Least Squares Approximation:
69Properties of Composition of Linear Transformations:: Algorithms and Pseudocode:
231Nonlinear Least Squares Approximation:
703.6 Invertible Linear Transformations
23213.3 Numerical Differentiation and Integration
71Properties of Invertible Linear Transformations:: Algorithms and Pseudocode:
233Numerical Differentiation:
72Eigenvalues and Eigenvectors
234Forward Difference Formula:
734.1 Characteristic Equation: Properties of the Characteristic Equation:
235Backward Difference Formula:
744.2 Eigenspaces and Geometric Multiplicity: Properties of Eigenspaces and Geometric Multiplicity:
236Central Difference Formula:
754.3 Diagonalization
237Numerical Integration:
76Conditions for Diagonalizability:: Properties of Diagonalizable Matrices:
238Trapezoidal Rule:
774.4 Complex Eigenvalues: Properties of Complex Eigenvalues and Eigenvectors:
239Simpson’s Rule:
784.5 Applications in Physics
240Gaussian Quadrature:
79Inner Product Spaces
24113.4 Ordinary Differential Equations
805.1 Definition and Properties: Properties of Inner Product Spaces:
24213.4.1 Initial Value Problems
815.2 Cauchy-Schwarz Inequality: Applications and Consequences:
24313.4.2 Euler’s Method
825.3 Orthogonal Vectors and Subspaces
24413.4.3 Runge-Kutta Methods
83Properties of Orthogonal Vectors:
24513.4.4 Adaptive Step Size Control
84Orthogonal Subspaces:
24613.4.5 Stiff ODEs
85Orthogonal Basis and Orthonormal Basis:: Gram-Schmidt Orthogonalization Process:
24713.4.6 Example: Radioactive Decay
865.4 Gram-Schmidt Process
24813.4.7 ODEs in Physics Examples: 13.5 Partial Differential Equations
875.5 Orthogonal Complements: Orthogonal Projections:
24913.5.1 Basic Concepts
885.6 Least Squares Approximation
25013.5.2 Finite Difference Methods
89Properties and Applications:: Geometric Interpretation:
25113.5.3 Stability and Convergence
90Analytic Geometry in 2D
25213.5.4 Finite Element Methods
916.1 Cartesian Coordinate System
25313.5.5 PDEs in Physics Examples
92Definition and Components:
25413.5.6 Example: 2D Laplace Equation: 13.6 Error Analysis
93Properties and Conventions:: Applications:
25513.6.1 Machine Precision
946.2 Lines and Equations
25613.6.2 Truncation Error
95Slope-Intercept Form:
25713.6.3 Error Propagation
96Properties:: Point-Slope Form:
25813.6.4 Precision vs Accuracy
976.3 Conic Sections
25913.6.5 Verification and Validation
98Circles:
26013.6.6 Example: Matrix Condition Numbers
99Ellipses:
261Special Functions
100Parabolas:: Hyperbolas:
26214.1 Gamma and Beta Functions
1016.4 Parametric Equations
26314.1.1 Gamma Function
102Definition and Representation:
26414.1.2 Gamma Function Numerical Evaluation
103Properties and Advantages:: Calculus with Parametric Equations:
26514.1.3 Beta Function
1046.5 Polar Coordinates
26614.1.4 Use in Physics
105Definition and Representation:: Conversion between Cartesian and Polar Coordinates:
267Some key examples:
1066.6 Transformations in 2D
26814.2 Legendre Polynomials
107Types of Transformations:: Compositions and Inverse Transformations:
26914.2.1 Definition and Properties
108Analytic Geometry in 3D
27014.2.2 Legendre Polynomial Expansions
1097.1 Cartesian Coordinate System
27114.2.3 Numerical Computation
110Definition and Components:: Properties and Conventions:
27214.2.4 Physics Applications
1117.2 Lines and Planes
273Electrostatics examples:
112Lines:: Planes:
274Quantum theory examples:: Relativity and cosmology examples:
1137.3 Quadric Surfaces
27514.3 Bessel Functions
114Spheres:
27614.3.1 Bessel’s Equation
115Ellipsoids:
27714.3.2 Bessel Functions of the First Kind
116Paraboloids:
278Some key properties:
117Hyperboloids:
27914.4 Spherical Harmonics
118Cones:
28014.4.1 Definition
1197.4 Cylindrical and Spherical Coordinates
28114.4.2 Key Properties
120Cylindrical Coordinates:: Spherical Coordinates:
28214.4.3 Numerical Computation
1217.5 Vector Functions and Space Curves
28314.4.4 Physics Applications
122Vector Functions:: Space Curves:
284Quantum mechanics examples:
1237.6 Transformations in 3D
285Electrostatics/Magnetostatics examples:: Gravity and general relativity:
124Translation:
28614.5 Hermite Polynomials
125Rotation:
28714.5.1 Definition and Properties
126Scaling:: Composite Transformations:
28814.5.2 Hermite Polynomial Applications
127Partial Derivatives
289Statistics/Probability Applications:: Physics Applications:
1288.1 Functions of Several Variables
29014.5.3 Numerical Computation : 14.6 Applications in Physics
129Graphical Representation:: Domain and Range:
29114.6.1 Quantum Theory
1308.2 Limits and Continuity
29214.6.2 Classical and Statistical Physics
131Limits:
29314.6.3 General Relativity and Cosmology
132Continuity:: Properties of Continuous Functions:
29414.6.4 Computational Methods
1338.3 Partial Derivatives
295Tensors
134Partial Derivatives:
29615.1 Definition and Properties
135Higher-Order Partial Derivatives:: Notation and Computation:
29715.1.1 What is a Tensor?
1368.4 Chain Rule
29815.1.2 Transformation of Tensor Components
137Statement of the Chain Rule:: Proof and Intuition:
29915.1.3 Symmetry Properties
1388.5 Directional Derivatives
30015.1.4 Tensor Algebra
139Definition:: Relation to Partial Derivatives:
30115.1.5 Tensor Fields: 15.2 Tensor Algebra
1408.6 Gradient and Tangent Planes
30215.2.1 Tensor Addition and Scalar Multiplication
141Gradient:: Tangent Planes:
30315.2.2 Tensor Products
142Multiple Integrals
30415.2.3 Contractions
1439.1 Double Integrals
30515.2.4 Quotients and Tensor Derivatives
144Definition and Notation:
30615.2.5 Example: Elasticity Tensor: 15.3 Tensor Calculus
145Geometric Interpretation:: Evaluation Methods:
30715.3.1 Covariant Differentiation
1469.2 Triple Integrals
30815.3.2 Gradient, Divergence, and Curl
147Definition and Notation:
30915.3.3 Curvature and Field Equations
148Geometric Interpretation:
31015.3.4 Example: Ricci Scalar Calculation: 15.4 Covariant and Contravariant Tensors
149Evaluation Methods:: Iterated Integral Method:
31115.4.1 Covariant and Contravariant Vector Components
1509.3 Change of Variables
31215.4.2 General Tensors
151Change of Variables in Double Integrals:: Change of Variables in Triple Integrals:
31315.4.3 Metric Tensors
1529.4 Applications in Physics
31415.4.4 Index Positioning Conventions: 15.5 Tensor Fields
153Mass and Center of Mass:
31515.5.1 Examples of Tensor Fields
154Potential Energy and Gravitational Fields:
31615.5.2 Tensor Field Transformation Properties
155Quantum Mechanics:: Fluid Dynamics and Heat Transfer:
31715.5.3 Tensor Fields on Curved Manifolds: 15.6 Applications in General Relativity
1569.5 Surface Integrals
31815.6.1 Differential Geometry Preliminaries
157Definition and Notation:: Parametric Surfaces:
31915.6.2 Einstein Field Equations
1589.6 Divergence and Curl
32015.6.3 Solving the Field Equations
159Divergence:
32115.6.4 Geometric Unitys Principle
160Curl:: Importance in Physics and Engineering:
322Glossary
161Vector Calculus
323Index
16210.1 Vector Fields
