Contents

Front matter

1. Preface
2. How to use this book

Part IStart here

How learning works, how to use AI without losing the skill, and how to plan a path that fits your week.

1. LMHow to learn: the session loop, error families, and AI without losing the skill5 exercises
2. Plan your path
3. Course coverage map
4. Prerequisite diagnostic and repair map16 exercises
5. Computing diagnostic and gap-specific repair map12 exercises

Part IIFoundations

Algebra, functions, and trigonometry, side by side with your first programs and proofs.

1. Module 01Algebra is the language of calculus8 exercises
2. Module 02Functions, inputs, domains, and graphs8 exercises
3. Module 03Composition, inverses, exponentials, and logarithms8 exercises
4. Module 04Radians, trigonometry, geometry, and units8 exercises
5. C01Programs as precise state changes5 exercises
6. C02Functions, recursion, search, and debugging5 exercises
7. C03C++ values, references, ownership, and lifetimes5 exercises
8. C04Logic, sets, relations, and constructing proofs5 exercises
9. C05Induction, counting, recurrences, graphs, and modular arithmetic5 exercises
10. Cumulative checkpoint: programming and discrete reasoning6 exercises

Part IIICalculus I

Rates, limits, derivatives, and integrals, each one computed, checked, and connected to code.

1. Module 05From two values to an average rate8 exercises
2. Module 06Limits: nearby behavior, not point substitution8 exercises
3. Module 07Continuity, existence, and precise limit reasoning8 exercises
4. Module 08The derivative and the tangent line8 exercises
5. W01From function inputs to executable contracts5 exercises
6. Mixed review R1: notation to tangent lines8 exercises
7. Module 09Derivative rules without losing the structure8 exercises
8. Module 10The chain rule and elementary derivatives8 exercises
9. Module 11Implicit, inverse, and logarithmic differentiation8 exercises
10. Module 12Motion, higher derivatives, and related rates21 exercises
11. Module 13Linear approximation, extrema, and the Mean Value Theorem22 exercises
12. Module 14Curve analysis and optimization21 exercises
13. Module 15Newton’s method, indeterminate forms, and growth8 exercises
14. Module 16Antiderivatives, Riemann sums, and definite integrals8 exercises
15. Module 17The Fundamental Theorem, substitution, and area8 exercises
16. Module 18Volumes from slices, washers, and shells8 exercises
17. Module 19Length, surface area, work, fluid force, and mass8 exercises
18. Cumulative assessment: Calculus I12 exercises

Part IVAlgorithms and systems

From data structures and proofs of correctness to machines, operating systems, databases, networks, and distributed systems.

1. C06Data structures, representation, and amortized cost5 exercises
2. C07Algorithm design, correctness, and complexity5 exercises
3. C08Trees, heaps, graphs, and shortest-path reasoning5 exercises
4. W02From invariants to algorithm design5 exercises
5. Cumulative checkpoint: algorithms and data structures6 exercises
6. C09Bits, instructions, caches, and virtual memory5 exercises
7. C10Operating systems, concurrency, and safe coordination5 exercises
8. W03From program objects to memory behavior5 exercises
9. C11Relational databases, transactions, and recovery5 exercises
10. C12Networks, byte streams, and protocol boundaries5 exercises
11. C13Partial failures, retries, and distributed consistency5 exercises
12. C14Software engineering, testing, debugging, and defensible measurement5 exercises
13. C15Languages, parsing, interpreters, and compiler structure5 exercises
14. C16Automata, decidability, reductions, and complexity classes5 exercises
15. W07The reach and limits of computational proof5 exercises
16. C17Security boundaries, usable interfaces, and professional responsibility5 exercises
17. Cumulative checkpoint: systems and engineering evidence6 exercises

Part VCalculus II

Integration strategy, honest numerical error, differential equations, series, and curves in the plane.

1. Module 20Integration by parts and structural choices8 exercises
2. Module 21Trigonometric integrals and trigonometric substitution8 exercises
3. Module 22Partial fractions and a mixed integration strategy8 exercises
4. Module 23Numerical integration and honest error bounds8 exercises
5. Module 24Improper integrals and probability density8 exercises
6. Module 25Introductory differential equations and dynamical models8 exercises
7. Mixed review R2: choosing a rate or accumulation method8 exercises
8. Module 26Sequences, series, and the meaning of convergence8 exercises
9. Module 27Positive series: comparison, integral, ratio, and root tests8 exercises
10. Module 28Alternating series, absolute convergence, and controlled error8 exercises
11. Module 29Power series and endpoint testing8 exercises
12. Module 30Taylor approximation, series, and remainder control8 exercises
13. Module 31Parametric curves and Cartesian conics8 exercises
14. Module 32Polar coordinates, polar calculus, and conics8 exercises
15. Cumulative assessment: Calculus II12 exercises

Part VICalculus III

Vectors, surfaces, gradients, multiple integrals, and the theorems of vector calculus.

1. Module 33Vectors, lines, planes, and surfaces in space8 exercises
2. Module 34Vector-valued motion, arc length, and curvature8 exercises
3. Module 35Multivariable functions, level sets, and limits8 exercises
4. Module 36Partial derivatives, differentiability, and the chain rule8 exercises
5. Module 37Gradients, directional derivatives, and tangent planes8 exercises
6. Module 38Multivariable extrema and constrained optimization8 exercises
7. Mixed review R3: approximation, geometry, and local change8 exercises
8. Module 39Double integrals, changing order, and polar area elements8 exercises
9. Module 40Triple integrals and cylindrical and spherical coordinates8 exercises
10. Module 41Jacobians and general changes of variables8 exercises
11. Module 42Vector fields, scalar line integrals, work, and circulation8 exercises
12. Module 43Conservative fields and Green’s theorem8 exercises
13. Module 44Surface area, scalar surface integrals, and flux8 exercises
14. Module 45Curl, divergence, and Stokes’ theorem8 exercises
15. Module 46The Divergence theorem and a unified view of calculus8 exercises
16. Cumulative assessment: Calculus III12 exercises

Part VIIApplied mathematics and computation

Linear algebra, probability, statistics, numerical methods, and differential equations, put to work in systems, GPUs, machine learning, robotics, and aerospace.

1. A01Linear systems, vector spaces, and transformations5 exercises
2. A02Orthogonality, least squares, eigenvectors, and SVD5 exercises
3. W04Geometry, gradients, and automatic differentiation5 exercises
4. A03Probability, random variables, and simulation5 exercises
5. A04Statistics, experiments, and uncertainty in measurements5 exercises
6. A05Numerical error, conditioning, roots, and approximation5 exercises
7. A06Differential equations, stability, and optimization5 exercises
8. W05From differential equations to trustworthy simulation5 exercises
9. T01Systems and performance engineering5 exercises
10. T02GPU, parallel, and graphics computing5 exercises
11. W06Linear algebra becomes graphics and spatial computation5 exercises
12. T03Machine learning, gradients, evaluation, and attention5 exercises
13. T04Embedded software, robotics, feedback, and state estimation5 exercises
14. T05Scientific and aerospace-related computing5 exercises

Part VIIIProjects

Five bounded engineering projects, each with a specification, independent tests, and stated limits.

1. J01Project 1: a tested telemetry analysis command1 exercise
2. J02Project 2: grid pathfinding with a correctness oracle1 exercise
3. J03Project 3: a transactional local reservation service1 exercise
4. J04Project 4: a numerical simulator with analytic validation1 exercise
5. J05Systems capstone: immutable spatial queries with verified parallel reads1 exercise

Part IXThe graduate core

Specifications, semantics, algorithms with certificates, measurement, architecture, and machine learning you can derive and defend.

1. GDGraduate-readiness diagnostic and repair map12 exercises
2. G01Specifications, invariants, and finite-state checking6 exercises
3. G02Language semantics, scope, and type safety6 exercises
4. G03Algorithm design, certificates, and honest complexity6 exercises
5. G04Probability, information, and credible measurement6 exercises
6. G05Numerical linear algebra and optimization6 exercises
7. G06Scalable architecture and measured performance6 exercises
8. G07Network design, routing, and queueing6 exercises
9. G08Cloud systems, partial failure, and consistency6 exercises
10. G09Systems analysis, contracts, and architecture decisions6 exercises
11. G10Project reasoning, dependency graphs, and risk6 exercises
12. G11Ethics, security boundaries, and accountable decisions6 exercises
13. G12Machine learning objectives, generalization, and decisions6 exercises
14. G13Neural computation and backpropagation you can derive6 exercises
15. G14Attention, embeddings, and transformer reasoning6 exercises
16. G15Data engineering, clustering, and defensible inference6 exercises
17. G16Research literacy, reproducibility, and independent engineering evidence6 exercises
18. GP1A typed compiler with a bytecode verifier1 exercise
19. GP2A reproducible model with retry-safe local delivery1 exercise
20. GACumulative reasoning and systems assessment6 exercises
21. GBCumulative learning, data, and research assessment6 exercises
22. W08Defending an independent engineering result5 exercises
23. From study outcomes to engineering evidence

Part XFrontier engineering

What engineers are building now, and the mathematics and code underneath it: drones that hold altitude, cars that follow a lane, and machines that run models fast.

1. F00The frontier map: what engineers build now, and your route in
2. F01Feedback control: derivatives that fly drones and steer cars12 exercises
3. F19Performance engineering on a parallel machine: bounds, VLIW packing, SIMD, and batched tree traversal10 exercises

Part XIThe AI Lab

From the chain rule to a small code model you train with reinforcement learning, reward with a grader you wrote, and then open up to see what it learned.

1. AI00The AI Lab route: from the chain rule to a code model you trained and opened up
2. AI01Autograd from scratch: backpropagation is the chain rule25 exercises

Part XIIThe Engineering Lab

From Newton’s second law to a drone you design, analyse, build, test, and fly with your own C++ attitude controller.

1. EL00The Engineering Lab route: calibrated to real jobs, built around one drone
2. EL01Why a drone hovers: forces, units, and your first design loop26 exercises

Part XIIIGeometry

From angle chasing to olympiad circles, with the two bridges an engineer needs: geometry as the constraint language of CAD, and geometry as code.

1. GEO00The Geometry route: seven phases, four bridges, one visual language
2. GEO01Angle chasing and your first proofs40 exercises

Part XIVThe Algorithms Lane

From your first loop to choosing, proving, and coding the right algorithm under a clock.

1. CP00The Algorithms Lane: fourteen modules, four builds, one habit
2. CP01Arrays, loops, and the frequency map15 exercises

References and tools

1. Reference sheets: formulas with conditions
2. Companion workflow and compact engineering references
3. Glossary: objects, guarantees, and common confusions
4. Theory map: the object, the guarantee, and the boundary
5. References: calculus
6. References: computing, mathematics, and careers
7. References: advanced and connecting work
8. How this book was checked
9. About the author

Also here

1. Index of terms
2. Interactive explorers
3. From the engine: the author’s code, chapter by chapter25 notes
4. Lab code: starters, tests and worked solutions