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

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

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.

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]

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.

Scope

The spine deliberately stops at the Riemann/Jordan level of integration: that is all the path to Stokes' theorem requires. Lebesgue measure and integration are out of scope (a possible future measure-theory.md); only forward pointers appear here. Differential geometry beyond what the integral theorems need (Riemannian metrics, connections, curvature) lives with the physics sections.