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← the syllabus

university levels of education Calculus I — IV

Four courses in one page, from the first limit to the wave equation — handmade, no ads, no sign-ups. Eighty-eight lectures, three hundred and fifty-two worked examples with every step reasoned, and eighteen ways to study the same material. Start at whichever level you need; the whole thing is one file and you can move between courses at any time.

§ levels 6 live
  1. 001

    Calculus I

    Limits, Derivatives and the Integral. The foundations. Limits and continuity, the definition of the derivative and every rule for computing one, then what derivatives are for — related rates, linear approximation, curve sketching, optimisation and l'Hôpital's rule — and finally the integral, Riemann sums, the fundamental theorem, substitution and areas between curves. Four chapters, twenty-two lectures.

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  2. 002

    Calculus II

    Techniques, Series, Parametric and Polar. Integration by parts, trigonometric integrals and substitution, partial fractions, numerical methods and improper integrals, then the applications that need them — volumes, arc length, surface area, work, centres of mass and probability. Then the theory of infinite series: convergence tests, absolute and conditional convergence, power series, Taylor and Maclaurin expansions, and curves described parametrically or in polar coordinates. Five chapters, twenty-two lectures.

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  3. 003

    Calculus III

    Multivariable and Vector Calculus. Calculus in three dimensions. Vectors, dot and cross products, lines, planes and surfaces, vector functions and curvature; partial derivatives, tangent planes, the chain rule, directional derivatives and the gradient, optimisation and Lagrange multipliers; double and triple integrals in polar, cylindrical and spherical coordinates with the Jacobian; and vector fields, line and surface integrals, conservative fields, curl and divergence, ending at Green's, Stokes' and the divergence theorems. Four chapters, twenty-two lectures.

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  4. 004

    Calculus IV

    Differential Equations, Transforms and Fourier Analysis. Where calculus becomes a modelling language. Separable, linear, exact and Bernoulli first-order equations with direction fields and Euler's method; higher-order linear equations by undetermined coefficients, variation of parameters and Cauchy-Euler methods, with vibrations and resonance; Laplace transforms including step functions, impulses and convolution; systems, eigenvalues and phase-plane stability; power series and Frobenius solutions with Bessel and Legendre functions; and Fourier series applied to the heat, wave and Laplace equations. Five chapters, twenty-two lectures.

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  5. 005

    Calculus V

    Advanced Calculus and Complex Analysis. The capstone. The rigorous foundations first — suprema and completeness, Bolzano-Weierstrass, epsilon-delta limits, uniform continuity and the Riemann integral put on solid ground, uniform convergence and the Weierstrass M-test. Then complex analysis from the plane and polar form through the Cauchy-Riemann equations, branches of the logarithm, contour integrals and Cauchy's theorem, the integral formula, Taylor and Laurent series, to the residue theorem evaluating real integrals. Then the Jacobian and the inverse and implicit function theorems, the Gamma and Beta functions, the Gaussian integral and differentiation under the integral sign, Fubini and Tonelli, and Laplace's method up to Stirling's formula. It ends with the calculus of variations: the Euler-Lagrange equation, the brachistochrone and the cycloid, geodesics, and Hamilton's principle with Noether's theorem. Four chapters, twenty lectures.

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  6. 006

    The whole of calculus

    All five courses in one reader — one hundred and eight lectures across twenty-two chapters, from the first limit to the calculus of variations. Every lecture explains its topic in full: sections of prose with the mathematics typeset by KaTeX, a table of the formulas that matter, four worked examples with the reason for every step spelled out, practice with hints and answers, a quiz, defined terms, and three problems left unsolved for the reader to do on their own. Nineteen ways to study the same material, from a plain reading mode and a continuous view that carries the whole subject end to end in one scroll, to drills, quizzes, flashcards, formula sheets, a searchable index, progress tracking and print. Forty-six colour palettes, twelve layouts and custom colours throughout.

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