Limits, continuity and infinite processes made rigorous.
12 topics · 15 curated works
No prior grounding assumed.
Essence of calculus
Grant Sanderson (3Blue1Brown) · 2017
Derives the rules of calculus from pictures rather than presenting them as formulae to memorise, so the chain rule and the fundamental theorem arrive…
Assumes you know the vocabulary.
Elementary Differential Equations with Boundary Value Problems
William F. Trench · 2000
Treats existence, uniqueness and stability as the organising questions of an ODE course, working systematically from first-order equations up to…
+3 more at this level
Primary sources and full treatments.
Intégrale, longueur, aire
Henri Lebesgue · 1902
Introduces the Lebesgue measure and integral, extending integration to a far larger class of functions than Riemann's construction allowed and…
+9 more at this level
12 of 15 works
Grant Sanderson (3Blue1Brown)
Derives the rules of calculus from pictures rather than presenting them as formulae to memorise, so the chain rule and the fundamental theorem arrive as things you could have invented.
link checked 17 Sept 2026William F. Trench
Treats existence, uniqueness and stability as the organising questions of an ODE course, working systematically from first-order equations up to boundary value and eigenvalue problems.
link checked 17 Sept 2026Matthias Beck, Gerald Marchesi, Dennis Pixton & Lucas Sabalka
Develops complex analysis from the Cauchy-Riemann equations onward, arguing that a function's complex differentiability is such a strong condition that it forces almost everything else to follow.
link checked 17 Sept 2026William F. Trench
Builds single-variable and then multivariable real analysis from an explicit axiomatic account of the real numbers, treating completeness as the property that makes calculus rigorous rather than assumed.
link checked 17 Sept 2026Grant Sanderson (3Blue1Brown)
Presents the Fourier transform as winding a signal around a circle and finding its centre of mass, which makes the frequency-domain view geometric rather than algebraic.
link checked 17 Sept 2026Henri Lebesgue
Introduces the Lebesgue measure and integral, extending integration to a far larger class of functions than Riemann's construction allowed and founding modern measure theory.
128 pageslink checked 17 Sept 2026David Hilbert
Extends the theory of quadratic forms to infinitely many variables and proves a spectral theorem for symmetric integral operators, laying the operator-theoretic groundwork Hilbert space would later be built on.
link failing as of 17 Sept 2026John von Neumann
Proves the mean ergodic theorem, showing that time averages of an observable along a measure-preserving flow converge in the mean, giving ergodic theory its first rigorous general result.
6 pageslink checked 17 Sept 2026Karl Petersen
Presents measure-preserving transformations, recurrence and entropy as one continuous story about how deterministic systems can behave statistically like random ones.
link checked 17 Sept 2026Brian S. Thomson, Judith B. Bruckner & Andrew M. Bruckner
Covers the standard first course in real analysis but insists on constructing the real numbers and proving convergence results from first principles before any calculus is assumed.
link checked 17 Sept 2026Terence Tao
Builds Lebesgue measure and integration from Jordan and Riemann measure outward, treating the failure of the simpler theory to handle countable operations as the motivation for every later construction.
link checked 17 Sept 2026Lloyd N. Trefethen
Argues Chebyshev interpolation and best polynomial approximation are best understood computationally, illustrating each theorem with a runnable numerical example rather than a stand-alone proof.