Lecture notes on the webMatthew McMillan
Semester courses
Material offered on this page is housed in a library where each topic is associated to a set of theory notes, example problems, exercises, and homework problems. The theory notes and exercises are collected into chapter files. Links to chapter file pages are given below, together with links to sections of those pages.
Calculus
Chapter 01
Chapter 02
Chapter 03
Chapter 04
Chapter 05
Chapter 06
Integration Techniques
Substitution
Trig power product: sin/cos · Swap an even bunch · u-sub for power-one · Power-to-frequency conversion · Trig power product: tan/sec or cot/csc · Special integrals: tan and sec
Partial fraction decomposition · Linear power bottom · Quadratic bottom, constant top · Quadratic bottom, linear top
Trapezoid rule - area formula · Midpoint Rule - area formula · Simpson’s Rule - defining formula · Simpson’s Rule - computing formula · Simpson’s Rule - error bound
Comparison test - integrals · p-integral cases · Additional improper integral types
Chapter 07
Integration Applications
Volume by shells - general formula
Arc-length formula
Area of revolution formula - thin bands
Pressure function · Fluid force on submerged plate · Fluid force for tilted surface
Moment · Centroid · Center of mass from moments · Midpoint of strips for opposite variables
Chapter 08
Sequences and Series
8.3Convergence
Monotone sequences · Series convergence
Simple Divergence Test (SDT)
Integral Test (IT) · p-series · Direct Comparison Test (DCT) · Limit Comparison Test (LCT)
Alternating Series Test (AST) - “Leibniz Test”
Ratio Test (RaT) · Root Test (RooT)
Chapter 09
Chapter 10
Parametric and Polar Curves
Parametric curve · Image of a parametric curve
Pure vertical, Pure horizontal movement
Arclength formula · Distance function · Speed function · Surface area of a surface of revolution: thin bands
Converting Polar ↔ Cartesian
Polar arclength formula · Sectorial area from polar curve · Sectorial area between polar curves
Chapter 11
Complex Numbers
Complex conjugate
All complex roots
Arithmetic · Exponents · Polar format · Conjugate · Division · Roots
Chapter 12
Curves in the Plane
12.1Introduction
Electricity and Magnetism · Fluid Dynamics · Plasma Physics - Fluids + EM Fields · Mathematics - Differential Geometry · Optimization
Parametric curve
Distance traveled · Length function · Speed · Curvature
Area under polar graphs
Chapter 13
Vectors and Surfaces
13.1Vectors
Vector space - addition and scaling · Components of vectors · Geometric vectors
Dot product · Cross product · Triple products
Lines and planes · Distances
13.4Quaternions
Spheres · Cylinders generalized · Ruled surfaces · Quadrics - ellipsoids · Quadrics - hyperboloids · Quadrics - paraboloids
Chapter 14
Space Curves
Derivative rules for parametric space curves · Tangent and normal vectors · Distance traveled, speed · Curvature · Acceleration decomposition
Arc length parametrization
Vector integrals · Position, velocity, acceleration · Kepler's Law of Ellipses
Chapter 15
Multiple Integrals
Rectangular domains · Non-rectangular domains · Volume under a surface · Order of integration
Area of a region
Density and total quantity · Average value · Center of mass · Moments
Chapter 16
Partial Derivatives and Optimization
Differentials · Differentiability
Chain rule over a curve · Chain rule over several parameters
Review of 1D theory · 2D theory · nD theory
Chapter 17
Line Integrals and Green's Theorem
17.1Vector fields
Gradient
17.3Curl
Curl as derivative operator · Curl as measure of circulation
Understanding Green's Theorem
Chapter 18
Flux, Divergence and Stokes' Theorem
18.2Divergence
Divergence as a derivative operator · Divergence as measure of expansion
Change of variables · Surface integral - scalar · Surface integral - vector - flux across
Vector curl · Stokes' Theorem · Path independence · Surface independence
Differential Equations
Chapter 01
First-Order Equations
Concepts of differential equations
Direct integration · Separable equations - substitution · Linear equations - product rule · Exact equations - gradients
Homogeneous equations - conversion to separable · Integrating factors - conversion to exact · Reduction of order · Ad hoc transformations
Excursus - functions, derivatives, integrals · Initial value problems · Functional viewpoint and IVPs - pathological phenomena · Picard-Lindelöf Theorem - local well-posedness · Picard iteration
Chapter 02
Qualitative and Numerical Methods
Paths in the plane · Growth and decay · Autonomous equations
Euler's method · Heun's method · Runge-Kutta methods · Adams method - multistep - polynomial fitting · Errors · Adaptive methods · Stability and stiff problems
Chapter 03
Higher-Order Linear Equations
Second-order, constant coefficients
Discovering a second solution · Particular solution of inhomogeneous
Damped harmonic oscillator · Driven harmonic oscillator · Driven, damped harmonic oscillator · Series RLC circuit · MIT Mathlet links
Chapter 04
Series Solutions and Laplace Transforms
Review of power series · Power series solutions - basic technique · Second-order linear equations
Chapter 05
Linear Algebra Tutorial
Basic vector algebra · Spans · Matrices - linear actions on vectors · Determinant · Independence · Matrix algebra - composition as multiplication · Eigenvectors · Finding eigenvalues and eigenvectors · Matrix calculus
Chapter 06
Linear Systems
Systems from higher order equations · Homogeneous system
Eigenvector approach · Complex eigenvalues · Insufficient eigenvectors · Matrix exponential approach
Chapter 07
Fourier Series and PDEs
Fourier series basics · Fourier series - orthogonality relations
Eigenfunctions
Traveling waves · Standing waves
Linear Algebra
Chapter 01
Vectors in 2D and 3D
Vector space - addition and scaling · Components of vectors · Geometric vectors · Dot product · Geometry · Lines and planes using vectors · Spans · Independence · Distances
Chapter 02
Matrices in 2D and 3D
Matrices acting on vectors · Diagonal matrices - stretching · Shearing · Rotating · Projecting · Reflecting · Permuting
Matrix multiplication · Matrices acting on matrices · Matrix algebra · Eigenvectors, eigenvalues
Chapter 03
Generators, Independence and Orthogonality
Basics of nD vectors · Spans · Subspaces · Spans and subspaces · Dimension
Independence · Basis
Orthogonal bases · Projection · Gram-Schmidt process · Application - distance from point to subspace
Chapter 04
Reduction, Inversion and Determinants
Keys to LU factorization · Computing LU factorizations · Pivoting · Row-echelon form
Reduced row-echelon form · Preimage of a vector · Solving linear systems · Preimage of a matrix
Chapter 05
Theory of Linearity
Linearity and the meaning of matrix multiplication · Change of basis · Matrix of change of basis · Image, kernel, transpose, rank
Chapter 06
Inner Products and Spectrum
Orthogonal matrices · Symmetric matrices · Quadratic forms and symmetric matrices · Constrained optimization, min-max principle · Positive definite symmetric matrices · Inner products and symmetric matrices · Addendum - determinants and eigenvectors
Eigenvectors, eigenvalues - Part II · Eigenvectors vs. singular vectors · Defining the singular value decomposition · Computing the singular value decomposition
Chapter 07
Least Squares and PCA
Least squares · Linear regression for summation curves · Multiple regression · Penrose quasi-inverse
Defining PCA · Discussion · Applications of PCA
Probability and Statistics
Chapter 01
Probability Models and Conditioning
Events and outcomes – informally · Events and outcomes – mathematically · Notation · New events from old
Axioms of probability · Probability model · Inferences from Kolmogorov
Multiplication rule · Law of Total Probability: 2 cases · Law of Total Probability: n cases
Chapter 02
Chapter 03
Discrete Random Variables
Random variable · PMF · PDF · CDF · PMF from CDF
Bernoulli variable · Binomial variable · Geometric variable · Pascal variable · Uniform variable
Expected value · Variance · Calculating variance · Standard deviation
Chapter 04
Poisson Process and Continuous Variables
Poisson variable · Exponential variable · Erlang variable
Notation · Expectation of derived variables · Linearity of expectation · Variance squares the scale factor · Relating PDF and CDF
Chapter 05
Joint Distributions
Joint PMF and joint PDF · Marginal PMF, marginal PDF · Joint CDF
Independence using PMF and PDF
5.3Functions on two random variablesProblems: Functions on two random variables, Sums of random variables
PMF of (any) function of two discrete variables · CDF of (continuous) function of two continuous variables
Chapter 06
Covariance and Conditional Distributions
Expectation for a function on two variables · Expectation sum rule · Expectation product rule: independence
Covariance · Correlation coefficient
Conditional distribution - fixed event · Conditional distribution - variable event
Expectation conditioned by a fixed event · Expectation conditioned by a variable event · Expectation conditioned by a random variable
Chapter 07
Limit Theorems
Summation formulas: E[X] and Var[X]
IID variables · Standardization · Normal approximation of binomial · De Moivre-Laplace Continuity Correction Formula
Tail probabilities · Markov’s inequality · Chebyshev’s inequality · Sample mean and its variance
Law of Large Numbers (weak form)
Chapter 08
Testing and Estimation
Significance test · Significance level
Binary hypothesis test · MAP design · ML design · MC design
Mean square error (MSE) · Minimal MSE of X given Y · Linear estimator: Line of minimal MSE
Data Science
Chapter 01
Topological Spaces
Spaces, subspaces, connectedness · Metric, open balls, continuity · Compactness, homeomorphism, continuity, homotopy
Quotients, gluing · Embedding, limit points, closure, boundary
Chapter 02
Simplicial Complexes and Homology
Simplices and complexes · Subdivision and triangulation
Boundaries, chain complexes
Chapter 03
Complexes, Homotopy and Betti Numbers
Delta complexes · Nerve complex · Čech complex
Homotopy type and homology
Betti numbers, Euler Characteristic
Chapter 04
Maps and Variations of Homology
Simplicial maps, chain maps · Induced maps on homology
Relative homology · Cochain complex · Cohomology · Coefficients
Chapter 05
Manifolds and Morse Theory
5.1Manifolds
Topological and smooth manifolds
Local coordinates
5.3Morse theory
Gradient, critical points; Hessian, non-degenerate points · Morse functions on manifolds · Morse functions and topology
Chapter 06
Persistent Homology and Mapper
Gromov-Hausdorff distance · Dendrogram · Filtrations · Vietoris-Rips complex · Persistent homology, barcodes · Barcode basis
6.2Stability
Bottleneck distance · Wasserstein metrics
Several parameters · Zig-zag persistent homology
Clustering and Reeb graph
Chapter 07
Manifold Learning
PCA · MDS · Isomap manifold learning · Local Linear Embedding (LLE)
Laplace-Beltrami operator · Laplacian for graphs
7.3UMAP
UMAP process · Gradient descent