The School of Numbers & Logic · mathematics, pure and applied
Calculus & Analysis
The mathematics of change and accumulation, followed by the harder question underneath: why the methods are entitled to work.
Instantaneous change made computable — limits, the derivative, and the applications that made calculus indispensable.
Syllabus · 5 units · ~45 hours
Unit I — Limits
The tangent problem and the velocity problem · Limits numerically, graphically, and by algebra · One-sided limits and continuity · Limits at infinity and asymptotic behavior
Unit II — The Derivative
The derivative as a limit of slopes · The derivative as a function · Differentiability and where it fails · Interpreting derivatives: velocity, marginal cost, rates
Unit III — Rules of Differentiation
Power, product, and quotient rules · The chain rule · Derivatives of trigonometric, exponential, and log functions · Implicit differentiation
Unit IV — Applications
Related rates · Linear approximation · Optimization: the best of all feasible worlds · L'Hôpital's rule, used with restraint
Unit V — The Shape of a Graph
The mean value theorem · First and second derivative tests · Concavity and inflection points · Curve sketching as synthesis
Accumulation from the definite integral to Taylor series — the second half of the calculus bargain.
Syllabus · 5 units · ~45 hours
Unit I — The Integral
Area under a curve by Riemann sums · The definite integral and its properties · The fundamental theorem of calculus, both parts · Antiderivatives and the indefinite integral
Unit II — Techniques of Integration
Substitution · Integration by parts · Trigonometric integrals and substitutions · Partial fractions · Improper integrals
Unit III — Applications of Integration
Areas between curves · Volumes by slicing and by shells · Arc length and surface area · Work, average value, and probability densities
Unit IV — Sequences & Series
Sequences and their limits · Geometric series and the harmonic series · Convergence tests: comparison, ratio, integral, alternating · Absolute versus conditional convergence
Unit V — Power Series
Power series and radius of convergence · Taylor and Maclaurin series · Approximating functions and bounding the error · Series solutions in miniature
Calculus in two, three, and n dimensions — gradients, multiple integrals, and the three great theorems of vector calculus.
Syllabus · 5 units · ~45 hours
Unit I — Space & Surfaces
Vectors, dot and cross products · Lines, planes, and quadric surfaces · Vector-valued functions and curves in space
Unit II — Partial Derivatives
Functions of several variables and their graphs · Partial derivatives and the tangent plane · The gradient and directional derivatives · Optimization and Lagrange multipliers
Unit III — Multiple Integrals
Double integrals over general regions · Polar, cylindrical, and spherical coordinates · Triple integrals and applications: mass, centers, moments
Unit IV — Vector Fields
Vector fields, work, and line integrals · Conservative fields and potential functions · Green's theorem
Unit V — The Big Theorems
Surface integrals and flux · Stokes' theorem · The divergence theorem · One theorem in three costumes: the unified view
Equations whose unknowns are functions — solved, sketched, and put to work on cooling, orbits, circuits, and populations.
Syllabus · 4 units · ~36 hours
Unit I — First-Order Equations
What a differential equation asserts · Slope fields and qualitative reasoning · Separable equations: growth, decay, cooling · Linear first-order equations and integrating factors · The logistic equation
Unit II — Second-Order Linear Equations
Homogeneous equations with constant coefficients · The characteristic equation and its three cases · Springs and circuits: damped and driven oscillation · Resonance
Unit III — Systems
Systems of first-order equations · Phase portraits and equilibria · Predator–prey and competition models · Linearization and stability
Unit IV — Solving by Other Means
The Laplace transform and initial-value problems · Numerical methods: Euler and Runge–Kutta in outline · Knowing when to trust a numerical answer
The theorems of calculus rebuilt on rigorous ground — completeness, epsilon-delta, and the honest small print of the subject.
Syllabus · 5 units · ~50 hours
Unit I — The Real Numbers
Why the rationals are not enough · The completeness axiom and suprema · The Archimedean property and density · Countable and uncountable sets
Unit II — Sequences
The limit of a sequence, defined and defended · The monotone convergence theorem · Subsequences and the Bolzano–Weierstrass theorem · Cauchy sequences
Unit III — Continuity
Epsilon-delta continuity, read slowly · The intermediate value theorem, proved · The extreme value theorem · Uniform continuity and why the distinction matters
Unit IV — Differentiation
The derivative from the definition · The mean value theorem and its consequences · Pathologies: continuous everywhere, differentiable nowhere
Unit V — Integration
The Riemann integral, constructed · Which functions are integrable · The fundamental theorem of calculus, earned · Sequences of functions and uniform convergence