Course coverage map
This map shows where this book teaches the standard topics of common university courses. Use it to study alongside a course, to prepare for one, or to fill in what a course you took left out. The topic lists follow common syllabi and, for the computing courses, the knowledge areas of the ACM/IEEE-CS/AAAI curriculum guidelines [CS2023]. They are not any one institution’s syllabus, and course boundaries vary: applications of integration (Modules 18–19), first differential equations (Module 25), and parametric and polar curves (Modules 31–32) move between Calculus I, II, and III from one school to the next. Follow your syllabus, and use the identifiers to find each topic here.
In the table, a bare number such as (12) is a module; the other identifiers are explained in “How to use this book”. Read the listed chapters in the order given; each chapter’s opening box names its prerequisites. Where a course often includes a topic that this book leaves out or only introduces, the row ends by saying so; study that topic from a dedicated text.
| Course | Where this book teaches its standard topics |
|---|---|
| Calculus I | Algebra, functions, exponentials, logarithms, and trigonometry: Modules 01–04, as the diagnostic directs. Average rates and limits: 05–07. The derivative, its rules, the chain rule, and implicit, inverse, and logarithmic differentiation: 08–11. Applications: motion and related rates (12); linear approximation, extrema, and the Mean Value Theorem (13); curve analysis and optimization (14); Newton’s method, indeterminate forms, and growth (15). Integrals: antiderivatives, Riemann sums, and the definite integral (16); the Fundamental Theorem, substitution, and area (17). Often included: volumes (18); length, surface area, work, fluid force, and mass (19). Workshop W01, review R1, assessment CI. |
| Calculus II | Integration by parts (20); trigonometric integrals and substitution (21); partial fractions and a mixed strategy (22); numerical integration with error bounds (23); improper integrals and probability densities (24); first differential equations (25); sequences, series, and convergence tests (26–28); power series, and Taylor series with remainders (29–30); parametric curves, polar coordinates, and conics (31–32). Review R2, assessment CII. |
| Calculus III | Vectors, lines, planes, and surfaces (33); motion, arc length, and curvature (34); functions of several variables and their limits (35); partial derivatives, differentiability, and the chain rule (36); gradients, directional derivatives, and tangent planes (37); extrema and constrained optimization (38); double integrals and polar area (39); triple integrals in cylindrical and spherical coordinates (40); Jacobians and changes of variables (41); vector fields, line integrals, work, and circulation (42); conservative fields and Green’s theorem (43); surface integrals and flux (44); curl, divergence, and Stokes’ theorem (45); the Divergence theorem (46). Review R3, assessment CIII. |
| Discrete mathematics | Logic, quantifiers, sets, relations, and proof techniques (C04); induction, counting, recurrences, graphs, and modular arithmetic (C05); finite probability (A03); invariants and recurrences in algorithm design (W02). |
| Data structures and algorithms | Recursion and binary search (C02); arrays, linked lists, stacks, queues, hash tables, and amortized cost (C06); divide and conquer, greedy algorithms, dynamic programming, backtracking, and asymptotic analysis (C07); trees, heaps, graph search, shortest paths, spanning trees, and disjoint sets (C08); from invariants to algorithm design (W02); checkpoint CA. Further: maximum flow, minimum cut, and certificates (G03); a pathfinding project (J02); timed practice in the Algorithms Lane (CP00–CP01). |
| Computer systems and architecture | C++ values, ownership, and lifetimes (C03); integer and floating-point representation, Boolean circuits, instructions, assembly, linking, caches, and virtual memory (C09); memory behaviour from source to hardware (W03). Further: performance engineering (T01); GPUs and parallel computing (T02); VLIW packing and SIMD on a parallel machine (F19); scalable architecture and measured performance (G06). |
| Operating systems | System calls, scheduling, processes and threads, paging, files, races, mutexes, condition variables, atomics, and deadlock (C10); virtual memory and address translation (C09, W03); verified parallel reads (J05). Introduced but not developed: scheduling algorithms and file-system internals. |
| Databases | SQL queries, normalization, indexes and query plans, transactions, isolation, migrations, backup, and recovery (C11); a transactional service (J03); consistency under partial failure (G08); joins and data engineering (G15). Not developed: storage-engine internals such as B-trees and join algorithms. |
| Computer networks | Byte streams and framing, TCP and UDP, HTTP, TLS, and backpressure (C12); partial failure, retries, idempotency, clocks, consensus, and the CAP theorem (C13); addressing, longest-prefix routing, queueing, and Little’s law (G07); cloud systems and consistency (G08). Not covered: the link and physical layers. |
| Software engineering | Testing, debugging, profiling, benchmarking, version control, interfaces, and observability (C14); security, threat modeling, accessibility, and professional responsibility (C17); executable contracts (W01); specifications and invariants (G01); requirements and architecture decisions (G09); planning and risk (G10); ethics and accountable decisions (G11); five bounded projects (J01–J05); defending an engineering result (W08). |
| Theory of computation | Finite automata, regular expressions, context-free grammars, pushdown automata, decidability and the halting problem, reductions, P versus NP, and NP-completeness (C16); the pumping lemma, automaton equivalence, and the halting contradiction worked in full (W07); grammars, parsing, and interpreters (C15); semantics and type safety (G02); a typed compiler (GP1). Not developed: Turing machines as a formal model; decidability is argued with programs. |
| Linear algebra | Linear systems, elimination, span, independence, bases, rank, null space, matrix multiplication as composition, inverses, and determinants (A01); orthogonality, projection, least squares, eigenvectors, and the singular value decomposition (A02); vectors in space (33). Further: gradients and automatic differentiation (W04); graphics transformations (W06); numerical linear algebra and optimization (G05). |
| Probability and statistics | Conditional probability, Bayes’ theorem, independence, random variables, expectation, variance, distributions, and simulation (A03); confidence intervals, hypothesis tests, effect sizes, experimental design, regression, and the bootstrap (A04); probability densities (24). Further: information measures and credible measurement (G04); clustering and inference from data (G15). Not developed: Markov chains and other random processes. |
| Numerical methods | Floating-point error, conditioning and stability, root finding, finite differences, and interpolation (A05); Newton’s method (15); numerical integration with error bounds (23); Taylor remainders (30); Euler and Runge–Kutta methods and their stability (A06); trustworthy simulation (W05); a simulator validated against analytic solutions (J04); scientific computing (T05); least squares by orthogonal transformations (G05). |
| Differential equations | Separable and linear first-order equations, integrating factors, and modeling (25); Euler’s method, Runge–Kutta methods, stability, and second-order equations as first-order systems (A06); the harmonic oscillator and the stability of its numerical solution (W05); feedback control (F01, T04); a validated simulator (J04). Not covered: Laplace transforms and the classical solution methods for higher-order linear equations. |
| Machine learning | Logistic regression, gradients and gradient checking, backpropagation, regularization, evaluation and data leakage, embeddings, and attention (T03); automatic differentiation (W04); an autograd engine you write yourself (AI01); objectives, generalization, and calibration (G12); backpropagation derived (G13); attention and transformers (G14); clustering and data engineering (G15); a reproducible model (GP2). Not developed: tree ensembles, support-vector machines, and convolutional networks. |
Studying alongside a calculus course#
Start. Take diagnostic D, repair Modules 01–02 as it directs, and work through 05–08, repairing 03–04 when they arise. Then attempt R1 before reading its solutions.
Finish the first course in coherent blocks. Study differentiation structure (09–11), applications (12–15), and accumulation (16–19). Take the Calculus I assessment after practising those ideas on separate days, not right after reading their solutions.
Build the second course. Work integration methods and modeling (20–25), then R2. Study convergence and approximation (26–30), then parametric and polar calculus (31–32). Take the Calculus II assessment. Module 25 and the assessments’ differential-equation items are optional unless your course includes them.
Build the third course. Study spatial geometry and local change (33–38), then R3; integration over regions and solids (39–41); then fields and boundary theorems (42–46). Take the Calculus III assessment once you can draw regions and explain orientations unaided. If your course orders topics differently, follow it and jump by identifier.