Calculus and Analysis

A self-contained spine of real analysis built for one purpose: to develop calculus rigorously enough to prove the classical integral theorems — Green's, Stokes', and the divergence (Gauss) theorem — and to present them as special cases of the generalized Stokes theorem

$${\int }_{M}d\omega ={\int }_{\partial M}\omega .$$

A second, shorter arc (part E) develops measure theory and the Lebesgue integral, which the Riemann spine repeatedly needs to name but cannot define, and on which probability and the $L^p$ spaces of functional analysis are built.

These pages are companion material to the rest of the Mathematics section and the Physics sections, which take this machinery for granted. They reuse the point-set topology and manifold definitions of topology-manifolds.md rather than re-deriving them, and supply in return the integration theory that page omits.

Contents

A. One-variable foundations

1. The Real Numbers and Completeness — ordered fields, the least-upper-bound axiom, Archimedean property, density, nested intervals; the load-bearing assumption of analysis.
2. Sequences and Series — convergence, monotone and Cauchy criteria, Bolzano–Weierstrass, $\mathrm{limsup}/\mathrm{liminf}$, series tests, uniform convergence, power series.
3. Limits and Continuity — $\epsilon$–$\delta$ limits, the topological view, the intermediate and extreme value theorems, uniform continuity.
4. Differentiation — the derivative as best linear approximation, the Mean Value Theorem, Taylor's theorem, smoothness classes.
5. The Riemann Integral — Darboux sums, integrability, the Fundamental Theorem of Calculus, change of variables, integration by parts.

B. Multivariable analysis

6. Metric Spaces and ${\mathbb{R}}^n$ — metrics and norms, completeness, compactness (Heine–Borel), and the contraction mapping theorem.
7. Multivariable Differentiation — the total (Fréchet) derivative, Jacobian, chain rule, gradient, symmetry of mixed partials.
8. Inverse and Implicit Function Theorems — local invertibility, level sets as submanifolds, the regular value theorem.
9. Multiple Integrals — integration over boxes and Jordan-measurable regions, Fubini's theorem, the change-of-variables formula and the Jacobian determinant.

C. Classical vector calculus

10. Vector Calculus — scalar/vector fields, grad, div, curl, line integrals, surface integrals, flux and circulation.
11. The Classical Integral Theorems — Green's, Stokes', and the divergence theorem, with proofs.

D. Forms and the general theorem

12. Differential Forms and Exterior Calculus — the wedge product, the exterior derivative $d$ (= grad/curl/div), pullback, orientation, integration of forms; $d^2=0$.
13. The Generalized Stokes Theorem — ${\int }_{M}d\omega ={\int }_{\partial M}\omega$, with the four classical theorems as corollaries. The capstone.

E. Measure and Lebesgue integration

14. Measure Theory — $\sigma$-algebras, measures and countable additivity, the $\pi$–$\lambda$ uniqueness theorem, Carathéodory extension, and Lebesgue measure on ${\mathbb{R}}^n$; measurable functions.
15. The Lebesgue Integral — the integral built by slicing the range, the monotone, Fatou, and dominated convergence theorems, the proof of Lebesgue's criterion, Radon–Nikodym, and the $L^p$ spaces.
16. Product Measures and Fubini — $\mu \otimes \nu$, Tonelli and Fubini, the measure-theoretic change of variables, and infinite products.
17. Non-Measurable Sets — the Vitali construction, Banach–Tarski, Solovay's model, and why measure theory quantifies over a $\sigma$-algebra rather than over all subsets.

Dependency graph

graph TD
  R[1 real-numbers] --> S[2 sequences-series]
  R --> C[3 continuity]
  S --> C
  C --> D[4 differentiation]
  D --> I[5 riemann-integral]
  R --> M[6 metric-spaces]
  C --> M
  M --> MV[7 multivariable]
  D --> MV
  MV --> II[8 inverse-implicit]
  I --> MI[9 multiple-integrals]
  MV --> MI
  MI --> VC[10 vector-calculus]
  II --> VC
  VC --> IT[11 integral-theorems]
  MI --> IT
  MV --> DF[12 differential-forms]
  VC --> DF
  DF --> SG[13 stokes-general]
  IT --> SG
  TM[../topology-manifolds.md] -.reuse.-> MV
  TM -.reuse.-> DF
  SG --> EM[../../physics/SR/fields/electromagnetism.md]
  I --> ME[14 measure-theory]
  MI --> ME
  ME --> LI[15 lebesgue-integral]
  LI --> PM[16 product-measures]
  ME --> NM[17 non-measurable]
  LI --> PR[../11-probability/00-README.md]
  PM --> PR
  LI --> FA[../06-functional-analysis/01-normed-banach.md]

Reading order

Read top to bottom (1 → 13); each page lists its prerequisites and where it is used. Readers who only need a specific result can jump in via the contents above — every page is cross-linked to its dependencies.

Part E (14 → 17) is a second spine, not a continuation of the first: it needs only pages 1–5 (plus §1 of multiple-integrals.md for the elementary volume of a box) and can be read directly after them. It is the prerequisite for probability and supplies the $L^p$ spaces that functional analysis assumes.

Scope

The path to Stokes' theorem (parts A–D) deliberately stays at the Riemann/Jordan level of integration: that is all the classical integral theorems require, and nothing in parts A–D depends on part E. Lebesgue measure and integration are developed separately in part E, where the two IOUs left by the Riemann theory — the definition of "measure zero" and the proof of Lebesgue's criterion — are discharged. Differential geometry beyond what the integral theorems need (Riemannian metrics, connections, curvature) lives with the physics sections; probability, which is measure theory with $\mu (X)=1$, lives in 11-probability/.