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university levels of education Calculus IV IV

Calculus IV, Deep Mastery Series — handmade, no ads, no sign-ups. Every lecture is kept whole and shown whole: nothing is summarised, so what you see is the lecture as it was written. 257,640 words across 120 lectures, from advanced integration to applied and computational calculus. Twenty-one ways through the same text, forty-six palettes on a white default with dark a click away, twelve layouts and colours you can set yourself. The mathematics is typeset from the author's own LaTeX, so every formula renders as written. Lectures 78 and 87 and 94 and 104 and 116 are the one gap; they were never in the source text, and the shelf says so rather than pretending otherwise.

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  1. 001

    Lectures 1–10

    Lectures 1 to 10. Ordinary Differential Equations: Introduction, First-Order Differential Equations, Modeling, Existence & Uniqueness, and Fundamental Solution Techniques, First-Order Differential Equations: Homogeneous Equations, Bernoulli Equations, Logistic Growth, Mixing Problems, Orthogonal Trajectories, and Advanced Mathematical Modeling, Exact Differential Equations, Potential Functions, Integrating Factors, Conservative Fields, and Advanced Solution Techniques, Higher-Order Linear Differential Equations: Constant Coefficients, Characteristic Equations, Repeated Roots, Complex Roots, Euler–Cauchy Equations, and Reduction of Order, Nonhomogeneous Linear Differential Equations: Method of Undetermined Coefficients, Variation of Parameters, Resonance, Damped Oscillations, and Forced Vibrations and 5 more, ending on Fourier Transform: Derivation, Properties, Inverse Transform, Convolution, Sampling, Dirac Delta, and Applications. 8,369 words across 10 lectures.

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  2. 002

    Lectures 11–20

    Lectures 11 to 20. Partial Differential Equations I: First-Order PDEs, Classification, Boundary Value Problems, Separation of Variables, Heat Equation, Wave Equation, and Laplace's Equation, Advanced Partial Differential Equations: Method of Characteristics, D'Alembert's Formula, Green's Functions, Eigenfunction Expansions, and Nonhomogeneous Boundary Value Problems, Complex Analysis I: Complex Numbers, Analytic Functions, Cauchy–Riemann Equations, Complex Integration, Cauchy's Theorems, Laurent Series, Residues, and Applications, Complex Analysis II: Laurent Series, Singularities, Residue Theory, Contour Integration, Branch Cuts, and Applications, Advanced Complex Analysis, Special Functions, Integral Transforms, Green's Identities, and Course Synthesis and 5 more, ending on Dynamical Systems, Nonlinear Differential Equations, Stability Theory, Bifurcations, Chaos, and Strange Attractors. 9,461 words across 10 lectures.

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  3. 003

    Lectures 21–30

    Lectures 21 to 30. Perturbation Methods, Asymptotic Analysis, Multiple Scales, WKB Theory, Boundary Layers, and Singular Perturbations, Integral Equations, Green's Functions, Fredholm Theory, Volterra Equations, Eigenfunction Expansions, and Applications, Functional Analysis: Normed Spaces, Banach Spaces, Hilbert Spaces, Linear Operators, Hahn–Banach Theorem, Open Mapping Theorem, and Spectral Theory, Measure Theory and Lebesgue Integration: Sigma-Algebras, Measurable Functions, Lebesgue Measure, Convergence Theorems, $L^p$ Spaces, and Modern Integration Theory, Distribution Theory (Generalized Functions), Sobolev Spaces, Weak Derivatives, Weak Solutions of PDEs, Variational Methods, and Finite Element Foundations and 5 more, ending on Modern Nonlinear Analysis: Fixed Point Theorems, Bifurcation Theory, Nonlinear Operators, Topological Degree, Monotone Operators, and Applications to Nonlinear PDEs. 10,799 words across 10 lectures.

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  4. 004

    Lectures 31–40

    Lectures 31 to 40. Fractional Calculus: Fractional Integrals, Fractional Derivatives, Caputo and Riemann–Liouville Operators, Fractional Differential Equations, and Applications, Stochastic Calculus: Brownian Motion, Itô Integrals, Itô's Lemma, Stochastic Differential Equations, Martingales, and Applications, Geometric Measure Theory: Hausdorff Measure, Fractal Geometry, Rectifiable Sets, Currents, Minimal Surfaces, and the Calculus of Irregular Objects, Nonlinear Waves and Solitons: Wave Propagation, Conservation Laws, Korteweg–de Vries Equation, Burgers' Equation, Nonlinear Schrödinger Equation, and Inverse Scattering, Spectral Theory and Eigenvalue Problems: Sturm–Liouville Theory, Orthogonal Functions, Spectral Decomposition, Green's Functions, and Applications and 5 more, ending on Calculus on Metric Measure Spaces: Sobolev Spaces Without Coordinates, Weak Gradients, Dirichlet Forms, Cheeger Energy, Curvature-Dimension Theory, and Modern Analysis. 11,360 words across 10 lectures.

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  5. 005

    Lectures 41–50

    Lectures 41 to 50. Noncommutative Calculus and Noncommutative Geometry: Operator Algebras, Spectral Triples, Noncommutative Differentiation, Cyclic Cohomology, Index Theory, and Applications, Infinite-Dimensional Morse Theory and Floer Theory: Critical Points on Function Spaces, Morse Homology, Symplectic Action Functionals, Floer Homology, and Modern Geometry, Geometric Calculus of Gauge Fields: Connections, Yang–Mills Theory, Fiber Bundles, Gauge Transformations, Curvature Forms, and Modern Mathematical Physics, Geometric Quantization: From Classical Calculus to Quantum Mechanics Through Symplectic Geometry, Deformation Quantization: Star Products, Poisson Geometry, Formal Power Series, Kontsevich Quantization, and the Semiclassical Limit and 5 more, ending on Derived Differential Geometry and Modern Geometric Analysis: Derived Manifolds, Derived Intersections, Virtual Fundamental Classes, Moduli Spaces, and Applications. 12,183 words across 10 lectures.

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  6. 006

    Lectures 51–60

    Lectures 51 to 60. Nonlocal Calculus and Peridynamics: Integral Operators, Nonlocal Gradients, Fractional Models, Nonlocal PDEs, and Continuum Mechanics, Geometric Deep Learning and Differential Geometry: Manifold Learning, Graph Calculus, Graph Neural Networks, Laplace–Beltrami Operators, Discrete Exterior Calculus, and Modern Data Geometry, Information Geometry and Statistical Manifolds: Fisher Information, Riemannian Metrics, Natural Gradient, Divergences, and Modern Optimization, Optimal Transport and Wasserstein Geometry: Mass Transport, Monge–Kantorovich Theory, Wasserstein Spaces, Gradient Flows, and Modern Applications, Mean Field Games and Mean Field Control: Hamilton–Jacobi Equations, Fokker–Planck Dynamics, Nash Equilibria, and Large Population Optimization and 5 more, ending on Microlocal Analysis and Fourier Integral Operators: Wave Front Sets, Pseudodifferential Operators, Propagation of Singularities, and Modern Geometric Analysis. 12,785 words across 10 lectures.

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  7. 007

    Lectures 61–70

    Lectures 61 to 70. Symplectic Geometry and Hamiltonian Calculus: Canonical Transformations, Poisson Geometry, Hamiltonian Systems, Moment Maps, and Modern Mathematical Physics, Contact Geometry and Contact Hamiltonian Systems: Odd-Dimensional Geometry, Contact Forms, Reeb Dynamics, Legendrian Geometry, Thermodynamics, and Dissipative Mechanics, Infinite-Dimensional Lie Groups, Gauge Geometry, Fiber Bundles, Connections, Curvature, and Modern Geometric Field Theory, Index Theory and Global Analysis: Elliptic Operators, Fredholm Theory, Atiyah–Singer Index Theorem, Heat Kernels, Spectral Invariants, and the Topology of Differential Equations, Geometric Quantization and the Bridge from Classical to Quantum Mechanics and 5 more, ending on Floer Homology and Infinite-Dimensional Variational Calculus. 15,254 words across 10 lectures.

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  8. 008

    Lectures 71–80

    Lectures 71 to 80. Gauge-Theoretic Moduli Spaces: Yang–Mills Theory, Instantons, Donaldson Invariants, Seiberg–Witten Equations, and Four-Manifold Topology, Ricci Flow, Geometric Evolution Equations, Entropy Functionals, Surgery Theory, and the Proof of the Poincaré Conjecture, Kähler Geometry, Complex Differential Geometry, Calabi–Yau Manifolds, and Kähler–Ricci Flow, Geometric Invariant Theory, Moment Maps, Stability, and Canonical Metrics, Moduli Spaces, Deformation Theory, Derived Moduli, and Virtual Fundamental Classes and 4 more, ending on Gauge Theory, Yang–Mills Equations, Instantons, Seiberg–Witten Theory, and Four-Manifold Invariants. 28,948 words across 9 lectures.

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  9. 009

    Lectures 81–90

    Lectures 81 to 90. Chern–Simons Theory, Knot Invariants, Floer Homology, Topological Quantum Field Theory, and Quantum Topology, Symplectic Geometry, Hamiltonian Systems, Moment Maps, Symplectic Topology, and Mirror Symmetry, Contact Geometry, Geometric Mechanics, Integrable Systems, Symplectic Dynamics, and Modern Hamiltonian Theory, Poisson Geometry, Geometric Quantization, Deformation Quantization, Star Products, and the Classical–Quantum Correspondence, Microlocal Analysis, Pseudodifferential Operators, Fourier Integral Operators, Wavefront Sets, and Modern Analysis of Partial Differential Equations and 4 more, ending on Ricci Flow, Geometric Evolution Equations, Hamilton's Theory, Perelman's Entropy, Surgery, and the Proof of the Poincaré Conjecture. 25,716 words across 9 lectures.

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  10. 010

    Lectures 91–100

    Lectures 91 to 100. Mean Curvature Flow, Curve Shortening, Geometric Singularities, Self-Shrinkers, Monotonicity, and Level-Set Methods, Harmonic Maps, Harmonic Map Heat Flow, Energy Minimization, Bochner Methods, Bubbling, and Geometric Applications, Ricci Flow I — Geometry Evolution, Curvature Smoothing, Hamilton's Equation, Maximum Principles, and Geometric Analysis, Modular Tensor Categories, Anyons, Topological Quantum Computation, Chern–Simons Theory, and Quantum Topology, Quantum Groups, Quantum Affine Algebras, Yangians, Integrable Systems, Bethe Ansatz, and Exactly Solvable Models and 4 more, ending on Geometric Langlands Program, Hitchin Systems, Higgs Bundles, Perverse Sheaves, D-Modules, Representation Theory, and Mathematical Physics. 36,229 words across 9 lectures.

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  11. 011

    Lectures 101–110

    Lectures 101 to 110. Factorization Algebras, Vertex Algebras, Conformal Field Theory, Operator Product Expansions, Chiral Algebras, and the Mathematics of Quantum Fields, Higher Topos Theory, ∞-Categories, Simplicial Sets, Quasi-Categories, and the Foundations of Modern Geometry, Motivic Homotopy Theory, $\mathbb A^1$-Homotopy, Motivic Spaces, Motivic Spectra, Algebraic Cobordism, and Motivic Cohomology, Derived Categories, Triangulated Categories, DG Categories, $A_\infty$-Categories, Stable $\infty$-Categories, and Homological Algebra, Higher Category Theory, Operads, Monoidal Categories, Infinity Operads, Higher Algebra, and Factorization Homology and 4 more, ending on Higher Category Theory, $(\infty,1)$-Categories, Quasi-Categories, Complete Segal Spaces, Higher Topoi, Derived Stacks, and Homotopy Type Theory. 31,785 words across 9 lectures.

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  12. 012

    Lectures 111–120

    Lectures 111 to 120. Derived Algebraic Geometry, Simplicial Commutative Rings, Cotangent Complexes, Derived Schemes, Derived Stacks, Spectral Algebraic Geometry, and Derived Moduli Problems, Microlocal Analysis, Pseudodifferential Operators, Fourier Integral Operators, Propagation of Singularities, Elliptic Theory, Index Theory, Heat Kernels, and Noncommutative Geometry, Motivic Homotopy Theory, $\mathbb A^1$-Homotopy, Motivic Cohomology, Algebraic $K$-Theory, Cobordism, Chromatic Homotopy Theory, Elliptic Cohomology, and Topological Modular Forms, Derived Algebraic Geometry, Spectral Algebraic Geometry, Higher Categories, ∞-Topoi, Derived Stacks, Factorization Homology, Topological Quantum Field Theory, and Modern Homotopical Geometry, Microlocal Analysis, Pseudodifferential Operators, Fourier Integral Operators, Propagation of Singularities, Index Theory, Atiyah–Singer Index Theorem, Spectral Geometry, and Modern Geometric Analysis and 4 more, ending on Information Geometry, Optimal Control, Hamilton–Jacobi Theory, Pontryagin's Maximum Principle, Differential Games, Stochastic Calculus, Malliavin Calculus, and Modern Mathematical Finance. 44,559 words across 9 lectures.

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  13. 013

    Lectures 121–125

    Lectures 121 to 125. Geometric Deep Learning, Information Theory, Neural Differential Equations, Physics-Informed Machine Learning, Scientific Computing, and Modern Computational Mathematics, Grand Synthesis of Advanced Calculus: Unifying Differential Geometry, Topology, Analysis, Probability, Dynamical Systems, Optimization, Mathematical Physics, and Modern Research, Research Frontiers in Modern Calculus: Fractional Calculus, Nonlocal Operators, Geometric Flows, Optimal Transport, Data-Driven PDEs, and Open Problems, Calculus on Manifolds: Exterior Calculus, Hodge Theory, de Rham Cohomology, Elliptic Complexes, and Global Analysis and Advanced Geometric Dynamics: Hyperbolic Geometry, Geodesic Flows, Anosov Systems, Symbolic Dynamics, Thermodynamic Formalism, Teichmüller Dynamics, Renormalization, and Complex Dynamics. 10,390 words across 5 lectures.

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  14. 014

    Lectures 78 and 87 and 94 and 104 and 116 — not supplied

    Lectures 78 and 87 and 94 and 104 and 116 are absent from the source text this course was assembled from. Every other lecture is here in full. Nothing needs rebuilding when the missing text turns up: adding it to calculus_4_lectures.json and rebuilding the helper file puts it on every shelf, in every mode and in the search index with no change to the page. This entry opens the index, where the whole course is laid out and each gap is a dashed cell.

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  15. 015

    The whole course

    All 120 lectures, opened without a filter. 257,838 words, 85,979 blocks, 192 tables and 0 paired examples, with a glossary of 2,200 entries and a search index of 6,506 words. Twenty-one ways through the same text. Six read it — one lecture entire, an index of every lecture number down the left, a reader that hands you a single section at a time, a continuous scroll, an outline of the section headings, and a syllabus of the whole course. Eight take it apart — every Spanish line beside its English, the glossary lifted out of the tables, the tables on their own, flashcards Spanish side first, the practice questions, a drill that pulls a random example in either direction, and the objectives and closing summaries. Seven keep track — homework, cultural notes, an accent-insensitive search across every block, starred lines, a page of your own notes per lecture, a progress board, and a printable handout. Forty-six colour palettes on a white default with dark one click away, eight colour roles editable by hand, twelve page layouts from a narrow reading column through ruled notebook paper and two magazine columns to a monospaced terminal, seven typefaces with a separate face for the Spanish, and sliders for text size, leading and line length. The English can be hidden to read the Spanish cold, or the Spanish hidden to work back the other way. Progress, stars and notes autosave, with JSON export and import.

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