A textbook in two volumes
Survey of Mathematics
University mathematics built from the ground up. Every term is defined before it is used, every proof can be stepped through one line at a time, and every chapter ends with practice that climbs from checking a definition to proving something new.
Volume 2, as a map
Each unit builds on the ones with arrows into it. Hover a unit to see what it rests on; open it for its story, its theorems in order, and a mixed review.
- 0Thinking in proofsHow to read a theorem, prove it, and check your own proof.
- 1What calculus really meansThe real numbers, limits, the derivative and the integral, made exact.
- 2Linear algebraVectors, linear maps and matrices: the language of almost everything.
- 3Linear algebra, deeperThe spectral theorem, the SVD, Jordan form, and the bridge to algebra.
- 4Space and closenessDistance, completeness, compactness: the bridge to analysis and topology.
- 5Differential calculus in many variablesThe derivative as the best linear approximation, in any dimension.
- 6Integration in many variablesLine, surface and volume integrals, and the theorems that connect them.
- 7Differential equationsChange described by equations, solved with linear algebra.
- 8Chance and computationProbability as measure, and how computers do mathematics.
Inside every chapter
- Worked examples you step through. One line at a time, with a "why?" behind each line and the answer at the end.
- Quick checks. Multiple choice and one-number answers, marked the moment you answer, with the reason.
- Proof workshops. Put a proof's lines in order, or choose the reason for each step.
- Pictures that move. Drag the points, slide the numbers, or press Play and watch the idea happen.
- Python labs. Short programs that run in the page, to compute what the chapter proves.
- Where it came from. Who found each idea, and the problem they were trying to solve.
The maths-major level
Proof, real analysis, linear algebra to the spectral theorem and the SVD, metric spaces, calculus in many variables, integration and Stokes' theorem, differential equations, probability and numerical methods.
65 chapters, open the contentsFoundations
For a first reader who knows the four operations and not much more: numbers, algebra, geometry and functions, up to the calculus that Volume 2 starts from, with the reasons school leaves out.
In preparationT taught in a course (MATH 375 or 376) A in Apostol's book, beyond the course B beyond both, added because later courses assume it or because it makes the rest make sense