Self-study curriculum · proposal · 2026-09-19

Mechanics to Relativity

Forty-one units from F = ma to Einstein's field equations, in which every piece of mathematics — vector calculus, ODE theory, Fourier, complex analysis, PDEs, variations, tensors, differential geometry — arrives late, at the exact moment a physical problem stalls without it. There is no unit called Vector Calculus, and there is no physics assumed anywhere.

41units

333 hslides + problems

6parts

24express route

35 + 6spine + branch

0fixed deadlines

The premise

Four commitments shape this curriculum. The first three are about what gets taught and in which order; the fourth is about whether the graph actually closes.

The physics leads and the mathematics follows, and there is no unit called “vector calculus”. You asked to start directly with mechanics, and unit 00 does: Newton's second law read as a differential equation and reduced to quadrature in the three cases where that works. None of vector calculus is assumed anywhere, because none of it is needed until unit 04, where the work integral demands it. Every tool then arrives under pressure with a problem already waiting for it — linear ODE theory with the damped driven oscillator, Frobenius and Bessel with Kepler's equation, Legendre with the Earth's oblateness, the calculus of variations with the brachistochrone, Fourier and Sturm–Liouville with the plucked string and the drum, complex analysis with causality, tensors with the inertia tensor, and differential geometry with the failure of flat spacetime to hold gravity. The tensor unit is called Rigid bodies and the PDE unit is called the wave equation. This costs nothing in rigour and buys the one thing that makes self-study of mathematical physics survivable: at every moment you know why you are learning this.

You have the maturity, so nothing is drilled, everything load-bearing is proved, and nothing is mimed. The course will not re-explain a limit, a linear map or an inner product. It will prove Picard–Lindelöf in unit 01, because determinism is a theorem and not a slogan; prove that the Euler–Lagrange equations are coordinate-invariant in unit 12, because that single fact is the seed of general covariance; prove Cauchy's tetrahedron theorem in unit 22, the one place in classical physics where you prove rather than assume that a physical quantity is bilinear; prove Noether in unit 13 so that unit 36's Killing vectors are recognisably the same theorem; and prove the Levi-Civita connection unique in unit 32 so that the Christoffel symbols are derived rather than posited. Mathematics carries the definition and physics carries the motivation, always in that order: a tensor is a multilinear map, and the transformation law is a theorem about its components under a change of basis. Every claim is either proved on the slide, proved in the notes, or labelled cited with where the proof lives — there is no fourth option, and where the proof is a research monograph (KAM, completeness of Sturm–Liouville eigenfunctions, Lovelock, the singularity theorems) the statement and its hypotheses are given exactly and the proof is cited rather than gestured at.

The three ways a mathematically strong beginner fails at physics are designed against explicitly. They are: no dimensional sense, no feel for approximation, and no picture of a system you cannot solve. So unit 00 proves the Buckingham π theorem and makes you predict the pendulum's period up to a function of one dimensionless group before you touch an integral; unit 21 is a full spine unit on asymptotics — optimal truncation, secular terms, boundary layers, Poincaré–Lindstedt — and it is the unit that sets up Mercury's precession as a secular term you deliberately do not remove; and unit 02 is a spine unit on phase portraits, Lyapunov functions and bifurcations, which feeds almost nothing downstream and stays on the spine anyway, because it is where a mathematician first sees a mechanical system rather than solving one. The reinforcement is numerical: almost every checkpoint ends in a quantity the world can refute. 31.8 h for the Foucault pendulum at the Panthéon, 98.6° for a sun-synchronous orbit at 800 km, 343 m/s for sound in air at 20 °C against Newton's wrong 290, 2.46 × 10⁻¹⁵ for Pound–Rebka, 42.98″ per century for Mercury. Dimensions and at least one limit are checked on every boxed result, and a derivation with the right structure and the wrong number has not finished.

It is a dependency graph, it is nearly a chain, and the express route is genuinely closed. Unlike the sister optimization course there are no three independent trunks here; each tool is load-bearing for the next piece of physics, and saying so is more honest than faking parallelism. What independence exists sits in six branch units — 15, 17, 34, 38, 39, 40 — and in one real fork: the analytic-and-field line (18–25) and the geometric line (16, then 26–33) both descend from unit 12 and rejoin at unit 28, which needs 27 from one side and 25 from the other. The graph was then built so that a short path to general relativity is a subgraph rather than a set of apologies: unit 26 derives the Lorentz group from homogeneity, isotropy and one invariant speed, with Maxwell (unit 25) earning that speed's value rather than defining it, and unit 35 gives two derivations of the field equations, the Bianchi–Lovelock–Newtonian-limit route needing only curvature and the equivalence principle. The express route is therefore 24 units with every needs satisfied on the route, and what it costs is named rather than hidden.

What a unit is

Every unit is the same four things. There is no lab and no code: the assessment is that the checkpoint is either true of you or it is not. A unit is done when the checkpoint is true — not when you have read the chapter.

Theory

A deck of 25–45 slides, speaker notes on every content slide, run with ./mr slides NN. Mathematics carries the definition, physics carries the motivation, and every claim is proved on the slide, proved in the notes, or labelled cited. Nothing is used before a unit in its needs has introduced it.

Worked examples

One running example per unit, introduced on the overview slide, returned to in every section and closed at the end, plus three or four fully worked examples carrying every step. Dimensions and at least one known limit are checked on every boxed result.

Problems and solutions

8–14 problems, at least one a proof rather than a computation: ./mr problems NN. Every problem has a full worked solution (./mr solutions NN) and a hint ladder that goes rung by rung from “what kind of object is the answer” to the first line (./mr hints NN). You cannot ask a question, so no step is “clearly”.

Checkpoint

One falsifiable claim about what you can now do, phrased so that failing it is unambiguous; usually it ends in a number the world can refute. It is the last slide and the head of the problem sheet, and ./mr done NN records it. If you cannot make it true, the unit is not finished — and you know precisely what is missing.

How long. Each unit's title slide carries its own estimate: 2–3.5 h of slides and 4–7 h of problems, so 6–10.5 h a unit. Summed over all 41 units that is 109 h of slides and 224 h of problems — 333 hours, plus 20–25 h for the capstone. By part: A 79.5 h, B 54 h, C 67.5 h, D 37.5 h, E 37.5 h, F 57 h. Take it as a scale, not a schedule; there are no deadlines in it. The CLI is standard-library Python, so there is no virtualenv to recreate and nothing to install — ./mr slides 00 works on a clone.

Who this assumes you are

Everything below is used hard and never re-taught. Everything not on this list is taught here, including all of the physics. To find out in an afternoon whether it holds for you, work the readiness sheet (./mr ready): a dozen problems on exactly this background, with hints and worked solutions, and a note on where to fill each gap. The part that decides is not proof but computational calculus — the worked solutions are long chains of integrals and series carried to a checked number.

Assumed

Mathematical maturity. You read and write proofs, you are comfortable with abstraction, quantifiers, ε–δ arguments and “suppose not”, and you habitually ask which hypothesis a theorem actually needs. Assumed everywhere; it is the reason the course can move this fast.
Linear algebra, properly. Vector spaces, bases, linear maps, dual spaces, determinants, eigenvalues, inner products, the spectral theorem for real symmetric matrices, Jordan normal form (stated, used in 06), quadratic forms and their signature, Sylvester's law of inertia. Units 07, 16, 26 and 31 cash this in hard.
Single-variable calculus and real analysis. Limits, continuity, uniform convergence, differentiation, the Riemann integral, integration by parts and substitution, Taylor with remainder, power series and radius of convergence.
Point-set topology of metric spaces. Open sets, compactness, completeness and the Banach fixed-point theorem — used in the first proof of the course (01) and again in unit 04's inverse function theorem.
Elementary group theory. Groups, homomorphisms, group actions. Used lightly for SO(3) in 16, one-parameter symmetry groups in 13 and the Lorentz group in 26. No representation theory assumed or taught.
A computer algebra system or numerical environment you are comfortable in. Nothing is built in this course, but units 09, 17, 19, 21, 37 and 39 have problems whose last step is arithmetic you should not do by hand.

Not assumed — all taught here

Any physics whatsoever. “Start directly with mechanics” means no pre-physics warm-up, not that anything may be assumed.
Dimensional analysis — unit 00, proved as a rank argument rather than absorbed by osmosis.
Multivariable and vector calculus — units 03, 04, 05.
ODE solution methods and qualitative ODE theory — units 00, 01, 02, 06, 09.
Fourier analysis — units 19 and 20.
Complex analysis — unit 20, as the tool that makes the Fourier transform usable.
Partial differential equations — unit 18, arriving as the continuum limit of a chain of oscillators.
Asymptotics and perturbation theory — unit 21, on the spine.
The calculus of variations — unit 11, conjugate points included.
Tensors — unit 16, as multilinear maps.
Differential forms — unit 34, as the unification and not as the entry point.
Differential geometry — units 31, 32, 33.

The shape of it

Solid arrows are the prerequisite a unit is built on — the one whose material the next deck opens by using, and it is always drawn however far away it sits. Dashed arrows are a unit's other needs, drawn only when the two units are near enough for the curve to say something: 14 of the 47 secondary prerequisites appear here, and the 33 long cross-part debts — unit 37 reaching back to 08, 10 and 21, unit 40 to 15, unit 34 to 24 — are left to each unit's card, which lists every need in full.

A · the equation of motion B · the action C · fields and the crisis D · Minkowski E · curved space F · Einstein secondary prerequisite

Part A — the equation of motion, and the calculus it forces

Newton's second law is a differential equation, and eleven units are one long answer to “which one, and how do you solve it?”

The mathematics arrives in five waves, each pulled in by a problem that stalls without it: dimensional analysis and quadrature for one-dimensional motion, qualitative theory for the equations with no formula, multivariable calculus for motion in space, line and flux integrals for work and gravity, and linear ODE theory plus series solutions for oscillators and orbits. It ends with the Kepler problem solved completely — shape and timetable — and the inclination of a real satellite predicted to within a degree. Eleven units, 79.5 h.

Newton's second law is an ODE — and dimensions tell you the answer's shape before you solve it

spine

needs nothing

Theory: The three laws and what each actually asserts — the first is the definition of an inertial frame, not a special case of the second, and the force law is constitutive input rather than theorem. One-dimensional motion under constant gravity, linear drag and quadratic drag; terminal velocity. Energy as a first integral, with turning points, bounded and unbounded motion and equilibria read straight off V.
Tools: Second-order ODEs reducible to quadrature in exactly three cases: F(t), integrate twice; F(v), separable then inverted; F(x), multiply by ẋ and integrate, which is where energy comes from. Separation of variables, implicit solutions, inverting t(x) and recognising when you cannot; improper integrals at a turning point and when the singularity is integrable. The Buckingham π theorem, stated and proved as a rank argument on the dimension matrix; similarity and scaling solutions. Elliptic integrals of the first kind named, not developed.
Problems: The slides do the plane pendulum twice on one slide: dimensionally, where the only groups available give T = √(l/g)·f(θ₀) with no mass dependence in two lines; then exactly, reducing to T = 4√(l/g)·K(sin(θ₀/2)) and finding the amplitude at which the small-angle formula is 1 per cent wrong (about 23°). The calculus determined f and nothing else. The sheet drills all three quadrature cases with the trap in each, then carries dimensional arguments to a number: Taylor's blast wave, R(t) = C(Et²/ρ)^1/5, read off the declassified Trinity photographs and recovering a yield near 21 kilotons — a classified number obtained from dimensions and a photograph — and the period of V = k|x|ⁿ as a Beta function, with the unique n for which period is amplitude-independent.; Checkpoint. Given any force of the form F(t), F(v) or F(x) you first write the answer's dimensional form and name the dimensionless group the calculus is left to determine, then reduce to quadrature and produce the explicit solution or the period integral — and you can classify which of the three forms a problem is in before starting it.

Determinism, and the one hypothesis it rests on

spine

needs 00

Theory: What “Newton's laws determine the future” means as a mathematical claim, and the two ways it fails. Norton's dome: a mass on a dome shaped so that it can spontaneously start moving after an arbitrary delay, with no force applied and no hypothesis of the second law violated. Finite-time blow-up: an inverse-cube attractor reaching the origin in finite time, and what that does to global determinism.
Tools: The phase-space reformulation: any mẍ = F(x, ẋ, t) is a first-order system on ℝ²ⁿ. Picard–Lindelöf proved via the Banach fixed-point theorem on C⁰ with a weighted sup norm, with the Lipschitz condition doing all the work. Gronwall's inequality and continuous dependence on initial data. Peano: existence without uniqueness. Maximal intervals of existence and blow-up.
Problems: The slides build Norton's dome in full: h(r) = −(2/3g)r^3/2 gives r̈ = √r; exhibit the one-parameter family r(t) = (t−T)⁴/144 together with r ≡ 0, verify every member solves the equation with identical initial data, and locate precisely which hypothesis fails — Lipschitz continuity at r = 0. The sheet verifies the hypotheses where you care and breaks them where you do not: prove the pendulum globally deterministic from local Lipschitzness on bounded sets; show ẋ = x² escapes in finite time and mẍ = −k/x³ reaches the origin in finite time, then decide whether either contradicts the theorem.; Checkpoint. You state and prove Picard–Lindelöf, and exhibit Norton's dome's one-parameter solution family together with the exact hypothesis that fails — without appealing to physical intuition at any step.

Phase portraits: what you know when there is no formula

spine

needs 01

Theory: The honest position, stated early because everything later depends on accepting it: almost no equation of motion has a closed-form solution, and that is not a gap in the literature. What you get instead is the qualitative picture — how many equilibria, of what type, with which basins, and whether anything periodic exists. The pendulum's separatrix on the cylinder. Damped systems as gradient-like flows and conservative systems as flows preserving a function, which is the first hint of the symplectic structure of unit 14.
Tools: Autonomous systems as vector fields; flows and the one-parameter group property. Equilibria, linearisation and the classification of planar linear systems by trace and determinant. Hartman–Grobman stated, with the sharp fact that it says nothing about centres. Lyapunov's direct method with LaSalle's invariance principle for the degenerate case. Limit cycles; Poincaré–Bendixson, stated with its essential reliance on the plane; the index of a vector field around a closed curve; the saddle-node, transcritical, pitchfork and Hopf bifurcations with normal forms.
Problems: The running example is the damped pendulum θ̈ + γθ̇ + sin θ = 0: the full cylinder portrait, the alternating saddles and spirals, then V = ½θ̇² + (1 − cos θ) and LaSalle proving asymptotic stability of the down position — a proof linearisation alone cannot give — and finally γ = 0, where the spiral becomes a centre, which is exactly the case Hartman–Grobman excludes. The sheet is about the traps: for ẋ = −y − x³, ẏ = x − y³ show linearisation predicts a centre while the origin is asymptotically stable and give the Lyapunov function that proves it; prove van der Pol has a limit cycle by constructing an explicit trapping region, without computing the cycle.; Checkpoint. Given a planar nonlinear system you locate and classify every equilibrium, decide whether linearisation is conclusive at each, construct a Lyapunov function where it is not, and either prove or rule out a periodic orbit using Poincaré–Bendixson together with the index theorem.

Where things are: curves, frames, and the frame that lies to you

spine

needs 00

Theory: Motion in three dimensions: projectile motion with drag, uniform circular motion and why the acceleration points inward. Rotating reference frames, with the centrifugal, Coriolis and Euler terms exposed as artefacts of differentiating in a frame that is itself turning. The Foucault pendulum, the eastward deflection of falling bodies, and why cyclones rotate.
Tools: Partial derivatives; the total derivative of a map ℝⁿ → ℝ^m as the linear map best approximating it, and the multivariable chain rule stated once as composition of linear maps rather than as a page of subscripts. Clairaut's theorem with its counterexample. Arc length and the Frenet–Serret frame, with a = v̇T + (v²/ρ)N. Time-dependent orthogonal matrices; antisymmetry of Ω = R^TṘ in 𝔰𝔬(3); the identification of 𝔰𝔬(3) with ℝ³ by the cross product; the transport theorem.
Problems: The slides derive the full rotating-frame equation by differentiating x_in = R(t)x_rot twice, then apply it to a pendulum at latitude λ to get a precession rate ω sin λ — 11.33° per hour at the Panthéon's 48.85° N, so a full turn in 31.8 h, against the observed roughly 32 h. The sheet applies the transport theorem until it stops being surprising: the eastward Coriolis deflection of a mass dropped 100 m at the equator (about 2.2 cm, with the sign right), and circular motion from scratch, showing ‖v‖ constant ⟺ a·v = 0 and deriving the Frenet decomposition from it.; Checkpoint. Starting only from x_inertial = R(t)x_rotating with R ∈ SO(3) you derive the Coriolis and centrifugal terms, identify ω from R^TṘ, and predict the Foucault period at a stated latitude to within an hour.

Work, and the arrival of vector calculus

spine

needs 03

Theory: Work as a line integral and the work–energy theorem. Conservative forces and potential energy; energy conservation as a first integral in three dimensions. Why friction and magnetism both fall outside this frame, for opposite reasons. Potential landscapes, with equilibrium and stability read off the Hessian.
Tools: Line integrals of vector fields; the gradient and the fundamental theorem of line integrals — with the distinction planted quietly and once: the derivative is a covector and the gradient is what you get after using an inner product to convert it, which is index-raising in unit 26 and nothing else. Green's theorem and Stokes' theorem in ℝ³, proved for a rectangle with the patching argument sketched. Curl as the local obstruction to being a gradient; the Poincaré lemma on a simply connected domain. Taylor in ℝⁿ, the Hessian and nondegenerate critical points; the inverse and implicit function theorems by contraction mapping; Lagrange multipliers derived from the implicit function theorem rather than asserted.
Problems: The slides show the inverse-square field conservative by constructing V = −GMm/r along a radial path, confirm path-independence on a deliberately awkward path, and then take the punctured-plane vortex F = (−y, x)/(x²+y²): curl zero everywhere it is defined, loop integral 2π, and a precise statement of what is being measured — the topology of the domain, not the field. The question is left open until unit 34. The sheet: five fields, three conservative and two not, each answered with V explicitly or a loop with nonzero circulation; and the monkey saddle V = x³ − 3xy², whose Hessian vanishes identically at its only critical point, so the classification has to come from the flow rather than from the second derivatives.; Checkpoint. Given F on an open set U you decide whether it is conservative and defend the answer by producing V explicitly or exhibiting a loop with nonzero circulation — including at least one case where the answer depends on the shape of U rather than on F.

Flux, Gauss, and gravity stops being a force between pairs

spine

needs 04

Theory: Universal gravitation and the shell theorem, which cost Newton years. The gravitational field as an object in its own right, with the field equation div g = −4πGρ, equivalently ∇²Φ = 4πGρ. Tidal forces as the variation of the field across a body, and the tidal tensor ∂_i∂_jΦ, trace-free in vacuum. Why the Moon raises two bulges and not one.
Tools: Multiple integrals and Fubini; change of variables and the Jacobian; spherical and cylindrical coordinates with their coordinate singularities identified. Surface integrals and flux; the divergence theorem, proved for a box with patching sketched. The Laplacian and harmonic functions; the mean-value property and the maximum principle. Solid angle, and why 1/r² is the unique power in three dimensions for which flux is radius-independent.
Problems: The slides prove the shell theorem twice — by brute integration over the sphere with the law of cosines, where the punchline hides in an algebraic cancellation, then by Gauss's flux law and symmetry in three lines — and compute Φ inside a uniform sphere, showing a body dropped down a tunnel through the Earth executes simple harmonic motion with period 84.3 min, wherever the tunnel's ends are. The sheet computes the tidal tensor of a point mass, shows the trace vanishes and the eigenvalues are (−2, 1, 1)GM/r³, and reads the two bulges off the eigenvectors — this object is the Newtonian limit of the Riemann tensor and unit 33 will say so — plus a non-symmetric distribution for which Gauss's law is true and useless, with a precise statement of what symmetry buys.; Checkpoint. You prove the shell theorem by Gauss's law in under half a page, and compute the tidal tensor of a point mass, showing it is trace-free and reading its two bulges off its eigenvectors.

The damped driven oscillator, which is most of linear physics

spine

needs 01 02

Theory: Small oscillations about a stable equilibrium as the universal local model: Taylor-expand any V at a minimum and the leading term is ½kx². Damping regimes and the quality factor Q; sinusoidal driving, resonance, the phase lag sweeping through it, the half-power width. The impulse response, and the claim that a linear system's response to everything is its response to one thing. Causality as a constraint on the response function — stated here, cashed at unit 20.
Tools: ℂ and the complex exponential as computational infrastructure — algebra, not yet analysis. Linear ODEs with constant coefficients: characteristic polynomial, the solution space as a two-dimensional vector space (Picard–Lindelöf from unit 01, not folklore), the Wronskian and Abel's identity, superposition, variation of parameters, resonance as the failure of the particular-solution ansatz. The Green's function of L = d²/dt² + γ d/dt + ω₀², built by matching a jump in the derivative across t = t′. Convolution and Duhamel's principle.
Problems: The slides solve ẍ + γẋ + ω₀²x = F₀ cos ωt completely — amplitude, phase lag sweeping 0 to π, peak position, full width at half power γ — then build G(t) from its defining property and recover the identical steady state by convolution: the same answer twice by unrelated machinery. The sheet: a square-pulse drive solved by convolution and separately by matching piecewise solutions at the pulse edges, agreeing at the corners; the amplitude response peaking at √(ω₀² − γ²/2) while the velocity response peaks exactly at ω₀, and which one a physicist means; and critical damping as a genuine Jordan-block degeneracy, the fastest return without overshoot.; Checkpoint. You construct G(t − t′) from its defining property, solve an arbitrary drive by convolution, verify against a directly computed piecewise solution, and can say why the solution space is two-dimensional by citing unit 01.

Many oscillators, and the spectral theorem earning its keep

spine

needs 06 04

Theory: Two masses and three springs; beats as interference between two normal modes. Normal modes and normal coordinates: the change of basis in which a coupled system becomes n uncoupled oscillators. N masses in a line with fixed ends; zero modes and the symmetries responsible for them; why a system with n coordinates has exactly n modes.
Tools: Linear systems ẋ = Ax; the matrix exponential; the general solution from eigendecomposition, with Jordan form for the degenerate case. The generalised eigenvalue problem Kv = ω²Mv with M symmetric positive definite, reduced by M^1/2 to a symmetric problem — hence real ω² and M-orthogonal modes, the spectral theorem doing physical work, and the reason the naive “diagonalise both” is false in general. Simultaneous diagonalisation of two quadratic forms when one is definite, with the counterexample when neither is. The Rayleigh quotient and the variational characterisation of the lowest frequency.
Problems: The slides take N equal masses and N+1 equal springs, try v_j = sin(jkπ/(N+1)) and get exactly ω_k = 2√(k/m) sin(kπ/2(N+1)); plotted at N = 20 the curve bends away from the straight line, and that bending is dispersion — the straight part is the sound wave unit 18 recovers in the continuum limit, and the correction −(ck)(ka)²/24 is a rare chance to see a continuum theory's own error bar. The sheet: beats in the two-mass system; verifying v_i^TMv_j = 0 on a chain with one heavy mass while Euclidean orthogonality fails; and bounding the lowest mode of a five-mass chain by a trial vector.; Checkpoint. For any (M, K) with M symmetric positive definite you produce the normal frequencies and mode shapes, verify orthogonality in the M-inner product rather than the Euclidean one, identify every zero mode with the symmetry responsible, and can explain why complex frequencies cannot occur.

The central-force problem, and the two conservation laws that solve it

spine

needs 04 06

Theory: Torque and angular momentum; conservation for central forces and the immediate consequence that motion is planar. The two-body problem reduced to one body of reduced mass μ. The effective potential V_eff = V(r) + L²/2μr², from whose graph the entire qualitative theory of orbits can be read — bound, unbound, circular, and the centrifugal barrier. Kepler's three laws derived rather than quoted. The Laplace–Runge–Lenz vector, and the fact that it is extra: most central forces have nothing like it.
Tools: The cross product and its identities. Conserved quantities as first integrals lowering the order of an ODE system, six to two here. Quadrature for r(t) and θ(r). The Binet substitution u = 1/r, turning the orbit into u″ + u = −(μ/L²u²)F(1/u) — linear for the inverse-square law, and the single most useful change of variable in the course, since unit 37 adds exactly one term to it and gets general relativity. Bertrand's theorem cited; its computational half is a problem.
Problems: The slides run Kepler end to end: from F = −k/r² to r = p/(1 + e cos θ) with p and e in terms of E and L, the classification by e, and T² = 4π²a³/GM from the areal velocity; then A = p × L − μkr̂, verified conserved by direct differentiation, with the orbit equation read off A·r in one line. The sheet: which power laws admit stable circular orbits (n > −3) and which close (only n = −2 and n = 1, from ω_θ√(3+n)); escape speed and the hyperbolic deflection angle, arriving at Rutherford's formula.; Checkpoint. From V(r) = −k/r alone you derive r(θ) = p/(1 + e cos θ), express e and p in terms of E and L, prove Kepler's second and third laws, and verify the Runge–Lenz vector is conserved by explicit differentiation.

Where is the planet at time t? Kepler's equation and the birth of Bessel functions

spine

needs 08

Theory: The orbit's shape was easy; its timetable is not. The eccentric anomaly and Kepler's equation E − e sin E = M: transcendental, with no closed-form inverse — the problem that occupied celestial mechanics for two centuries and that every ephemeris still solves.
Tools: Solution by series in a parameter; the Fourier–Bessel expansion E = M + 2∑J_n(ne) sin(nM)/n, which is where Bessel functions were actually born — its sine coefficients computed directly from the orthogonality of the sines, which is all this unit borrows, since unit 19 is where that becomes a completeness theorem. Bessel's equation; the Frobenius method at a regular singular point, the indicial equation, the two roots and the logarithmic second solution when they differ by an integer. Recurrence relations, the generating function, Bessel's integral representation, and the asymptotic √(2/πx) cos(x − nπ/2 − π/4). Lagrange inversion as the general mechanism. The Laplace limit e = 0.6627434…, asserted here and proved in unit 20 — a debt deliberately incurred.
Problems: The slides derive the Bessel series for E(M) by computing the Fourier coefficients of E − M and recognising Bessel's integral representation, then locate Mars (e = 0.0934) at a stated epoch to five significant figures with three terms, cross-checked against Newton's method on the transcendental equation with the iterations each route costs counted. The sheet is Frobenius practice and honest convergence: Bessel's equation of order zero by series with its logarithmic second solution, the method detecting the obstruction on its own; and Halley's comet at e = 0.967, past the Laplace limit, where the series fails numerically and you locate empirically where it begins to diverge.; Checkpoint. You derive the Bessel-function solution of Kepler's equation from its Fourier coefficients and use it to place a planet to five digits — and you can state exactly where it diverges while owing unit 20 the reason.

Real planets are not points: multipoles, Legendre, and secular perturbation

spine

needs 05 08

Theory: The potential of a non-spherical body. The Earth's oblateness and its J₂ = 1.0826 × 10⁻³, the largest non-spherical term in the solar system by effect, and its consequence: a satellite's orbital plane precesses, which is exploited rather than fought. Tides revisited as the ℓ = 2 term, and why the Moon is tidally locked.
Tools: The generating function (1 − 2xt + t²)^−1/2 defining the Legendre polynomials; Legendre's equation and its polynomial solutions; Rodrigues' formula; orthogonality on [−1, 1]. The multipole expansion of 1/|r − r′| with monopole, dipole and quadrupole terms and their coordinate-independent meanings. First-order perturbation theory for ODEs: osculating orbital elements, the method of averaging, and the distinction between secular and periodic terms — the first appearance of the idea unit 21 develops properly and unit 37 spends.
Problems: The slides expand Φ for an axially symmetric body, identify J₂ with the quadrupole moment, derive the nodal precession rate, and then solve Ω̇ = 360°/year for a circular orbit at 800 km altitude to get i = 98.6° — the actual inclination of every sun-synchronous satellite ever flown. The sheet: J₂ for an oblate spheroid of given flattening, checked against the Earth's f = 1/298.257; and the apsidal precession the same term produces, with the critical inclination 63.4° at which it vanishes — which is why Molniya orbits use exactly that.; Checkpoint. From J₂ alone you predict the sun-synchronous inclination at a stated altitude to within a degree of the published value, and you can say which Legendre term produces each of the two precessions.

Part B — the action, and coordinates ceasing to matter

Nothing here says anything Newton did not, and that is exactly the point

It is a change of language whose entire value is what it makes obvious. Extremising an integral gives the equations of motion; those equations are then invariant under any change of coordinates, which is the idea that ends the course as general covariance. Symmetry becomes conservation by one theorem. And the part quietly builds the two objects relativity needs — an action principle, and a real definition of a tensor, arriving with the inertia tensor because that is the first place nine numbers per basis will not do. Seven units, 54 h.

The brachistochrone, and how to differentiate with respect to a curve

spine

needs 04

Theory: Johann Bernoulli's 1696 challenge. Fermat's principle of least time, with Snell's law falling out of it. The hanging chain. The common structure: nature appears to extremise an integral, and the question is what equation that forces.
Tools: Functionals on spaces of curves; admissible variations and the first variation as a Gâteaux derivative. The fundamental lemma of the calculus of variations and the du Bois-Reymond form you actually need, both proved with bump functions. The Euler–Lagrange equation; the Beltrami identity when the integrand lacks explicit dependence on the independent variable; natural boundary conditions at a free endpoint; isoperimetric and holonomic constraints by Lagrange multipliers in function space; several dependent variables and higher derivatives. Then the second variation, the Legendre condition, the accessory equation and Jacobi's conjugate points — because “stationary action” is not “least action” and unit 26 will need to know which. The direct method, with Weierstrass's demolition of Dirichlet's principle as the reason existence is a real question.
Problems: The slides do the brachistochrone in full: set up the time functional, apply Beltrami, integrate to the cycloid in parametric form, fix both constants from the endpoints, and confirm with explicit numbers that it beats the straight line and the circular arc — then the catenary by the same route with the length constraint handled by a multiplier that is physically the tension, closing with a conjugate-point check that the cycloid really is a minimum. The sheet: Snell from Fermat, then its continuous version for n(y) solved in a linear gradient — a mirage, with a number; a great-circle arc exceeding half the circle, stationary but not minimal, with the conjugate point identified as the antipode; and geodesics on a surface of revolution, whose first integral is Clairaut's relation r sin ψ = const, which unit 36 re-derives as a Killing vector.; Checkpoint. You derive the cycloid from scratch, and you produce a functional whose Euler–Lagrange solution is stationary but not a minimum, together with the test — conjugate point or second variation — that detects it.

Lagrangian mechanics: coordinates stop mattering

spine

needs 11 08

Theory: Hamilton's principle with L = T − V; d'Alembert's principle and virtual work. Generalised coordinates, with holonomic constraints eliminated by choosing coordinates rather than by computing constraint forces, and the forces recovered by multipliers when you want them. The double pendulum; the bead on a rotating hoop and its bifurcation. Gauge freedom L → L + df/dt. Velocity-dependent potentials, with the Lagrangian for a charge in an electromagnetic field written down now and justified in unit 24. Non-holonomic constraints, with an honest statement that their variational treatment is subtle and often done wrong.
Tools: Proof that the Euler–Lagrange equations of L = T − V are equivalent to Newton's equations in Cartesian coordinates. Then the central proof: they are form-invariant under any invertible change of coordinates q → Q(q, t) — the reason the formalism exists, and the first appearance of the idea that ends the course as general covariance. Generalised momenta and cyclic coordinates; linearisation about equilibria, reconnecting with unit 07's generalised eigenvalue machinery unchanged; configuration space as a manifold, informally here and precisely in unit 31.
Problems: The slides take the bead on a hoop rotating at Ω: derive L, read off V_eff, and find the pitchfork bifurcation at Ω² = g/R where the bottom goes unstable and two equilibria appear at cos θ = g/RΩ² — three lines against the page a force balance takes, with unit 02's normal form recognisable in it and the phase portrait drawn on each side; then the double pendulum's L and its linearised normal modes. The sheet: a spherical pendulum treated in (θ, φ) and again in Cartesian coordinates with a multiplier, shown to agree; L and L + df/dt giving identical equations but different canonical momenta, so you name the quantity that is not gauge-invariant; and Atwood's machine with an elastic string, three degrees of freedom, where the point is that you never draw a free-body diagram or name a tension.; Checkpoint. You write L for one system in two genuinely different coordinate systems, derive both sets of Euler–Lagrange equations, show they describe the same motion, and state the theorem that guaranteed this in advance.

Noether: every conservation law you have met is one theorem

spine

needs 12

Theory: Momentum conservation is spatial homogeneity, angular momentum is isotropy, energy is invariance under time translation. The Galilean boost and its less familiar conserved charge. The Runge–Lenz vector of unit 08 finally explained as the charge of a symmetry that is not a symmetry of space at all — the hidden SO(4) of the Kepler problem. And what the theorem does not say: the converse is false.
Tools: One-parameter groups of transformations of configuration space, the induced variation of the action, and the infinitesimal generator. Statement and complete proof of Noether's theorem for point mechanics — three lines, which this learner will appreciate seeing — including the quasi-invariant case where L changes by a total time derivative. Why the energy function h = ∑q̇_i ∂L/∂q̇_i − L is the conserved quantity, and equals T + V only when the constraints are time-independent: a distinction almost universally glossed over, and one this course needs later.
Problems: The slides prove Noether in full generality and then run it five times on the Sun–Jupiter Lagrangian: time translation gives energy, spatial translation gives momentum, rotation gives all three components of angular momentum, the Galilean boost q → q + vt is quasi-invariant and gives G = pt − mq (the centre of mass moving uniformly, restated as a conservation law), and the SO(4) generator gives Runge–Lenz. The sheet: unit 12's bead on the rotating hoop, whose conserved h is not the energy — compute both and reconcile them with the work the motor does; the scaling symmetry (t, r) → (λ³t, λ²r), with Kepler's third law falling out of the charge; and a conserved quantity arising from no point symmetry, with the enlargement to phase space (unit 14) that would rescue it.; Checkpoint. Given any Lagrangian and any one-parameter symmetry you write the conserved quantity and verify it is constant on-shell — including one quasi-invariant case where L itself is not invariant and the charge acquires the extra term.

Hamilton: phase space, and flows that preserve volume

spine

needs 13 02

Theory: The phase-space picture and what it makes visible that configuration space hides: the pendulum's separatrix from unit 02 now as a level set of H, the structure of the Kepler problem, and the fact that trajectories never cross. Liouville's theorem and its meaning — a cloud of initial conditions deforms without shrinking, which is why statistical mechanics is possible and why integrators for mechanics should be symplectic.
Tools: The Legendre transform done properly as convex duality: definition by supremum, involutivity for convex functions, the degenerate case flagged in advance (unit 27's relativistic free particle is nearly one), and the relation to Young's inequality. Hamilton's equations as a first-order system on ℝ²ⁿ. Poisson brackets with bilinearity, antisymmetry and the Jacobi identity; {·, H} generates time evolution, so conserved quantities are exactly those commuting with H — Noether's converse, now cheap. Canonical transformations and the four generating functions; Liouville as divergence-freeness of the Hamiltonian vector field; the symplectic form ω = dp ∧ dq named here and unpacked in unit 34.
Problems: The slides Legendre-transform the Lagrangians of units 12 and 08, sketch the Kepler phase portrait in (r, p_r) at fixed L with the separatrix at E = 0, then compute {L_i, L_j}, {L_i, A_j} and {A_i, A_j}, watching the algebra close on 𝔰𝔬(4) for bound orbits and 𝔰𝔬(3,1) for unbound ones — which is what “hidden symmetry” meant in unit 13. The sheet: test a given transformation for canonicity three ways — M^TJM = J, preservation of all Poisson brackets, and an explicit generating function — and confirm the verdicts agree; and follow a rectangle of initial conditions through the free-particle and oscillator flows, computing the area at time t and watching the shear do all the work.; Checkpoint. You prove a given transformation canonical by two independent routes, perform a Legendre transform including its degenerate case, and prove Liouville's theorem from Hamilton's equations in three lines.

Hamilton–Jacobi: mechanics was optics all along, and free motion is already geodesic motion

branch

needs 14 11

Theory: Action as a function of its endpoints; solving a mechanical problem by solving one PDE and then differentiating. Action–angle variables: the frequencies of a bounded motion without ever solving for the motion. Adiabatic invariants via the slowly shortened pendulum. Then the two payoffs that make this the course's quiet centre. The optical–mechanical analogy — Fermat against Maupertuis, wavefronts against surfaces of constant S, rays against trajectories. And the Jacobi metric: a particle of energy E in a potential V moves on a geodesic of the conformally rescaled metric 2(E − V)δ_ij. Free motion is geodesic motion, in Newtonian physics, before any curvature exists. This is the branch that makes unit 32 feel inevitable rather than imposed, and the one worth fighting hardest to keep.
Tools: The Hamilton–Jacobi equation ∂S/∂t + H(q, ∂S/∂q, t) = 0, the course's first nonlinear first-order PDE. Complete integrals versus general solutions; Jacobi's theorem proved in the verification direction, which needs only the chain rule. Separability and Stäckel's conditions, stated. Action variables I_i as loop integrals of p dq over topological cycles, and the conjugate angles. The eikonal limit; conformal rescaling of a metric.
Problems: The slides solve Kepler by Hamilton–Jacobi separation in spherical coordinates, reading the orbit off ∂S/∂α = const without integrating an equation of motion, then compute the action variables, obtain H = −mk²/2I² and hence the Kepler frequency by differentiation — and note that all three frequencies being degenerate is exactly why Kepler orbits close, which is Bertrand's theorem seen from above. Then the Jacobi metric for the same problem, with Kepler orbits exhibited as its geodesics. The sheet: for V = ½kx² + λx⁴ compute I(E) to first order and get the amplitude-dependent frequency shift, checked against unit 21's Poincaré–Lindstedt answer; and adiabatic invariance for a slowly shortened pendulum, with the work done pulling the string through a hole computed two ways.; Checkpoint. You solve a separable Hamilton–Jacobi problem, extract action–angle variables, produce the oscillation frequency of an anharmonic oscillator without ever integrating its equation of motion, and write the Jacobi metric for a given V and E.

Rigid bodies, and what a tensor actually is

spine

needs 12 05 03

Theory: Rigid-body kinematics; the configuration space SO(3) × ℝ³. Angular momentum L = Iω, and the fact that L is generally not parallel to ω, which is the whole subject. Principal axes; the parallel-axis theorem; Euler's equations in the body frame; the free symmetric top and its two cones; the gyroscope; the intermediate-axis (tennis-racket) theorem. Euler angles and their coordinate singularities, which unit 31 will explain are a fact about SO(3) and not about carelessness.
Tools: Multilinear maps as the definition of a tensor: a (0,2)-tensor is a bilinear map V × V → ℝ, and the transformation law is a consequence of a change of basis rather than a definition. (p,q)-tensors, tensor products, contraction, symmetric and antisymmetric parts. The dual space and why it is not canonically V — the covector-versus-gradient point planted in unit 04, now made structural, so that units 28 through 35 can raise and lower honestly. Index notation and the Einstein summation convention introduced properly. The spectral theorem producing principal axes. SO(3), 𝔰𝔬(3) and the exponential map.
Problems: The slides compute the inertia tensor of a uniform rectangular block about a corner, diagonalise it, verify I′ = RIR^T component by component under a 45° rotation, then linearise Euler's equations about each principal axis and prove the intermediate axis unstable with a growth rate you can test with a book and a rubber band and check against the Dzhanibekov footage's flip period. The sheet: compute I in two frames and verify the transformation law, then exhibit a collection of nine numbers per basis that is not a tensor and say precisely what it fails; and derive the steady precession rate of a heavy symmetric top with its nutation, identifying which approximation the elementary torque-equals-L̇ argument silently makes.; Checkpoint. For an explicitly given body you compute I in two frames, verify the transformation law component by component, predict which axis is unstable, and can explain in one sentence why nine numbers per basis is not enough to be a tensor.

Integrability and chaos: determinism without predictability

branch

needs 14 02 01

Theory: The three-body problem and what Poincaré found in it, including the error in the prize memoir. Sensitive dependence; the double pendulum. Integrable systems as the exception rather than the rule, with Liouville–Arnold tori. KAM as the statement that most tori survive small perturbation and the ones that do not are exactly the resonant ones; resonances and the Kirkwood gaps in the asteroid belt, an observable consequence with a number attached. What all this implies for the stability of the solar system.
Tools: Complete integrability; the Liouville–Arnold theorem with hypotheses and the geometry of its proof. Poincaré sections and return maps; area-preserving maps; the standard map; Chirikov's overlap criterion. Lyapunov exponents defined via the linearised flow, with Oseledets cited. KAM stated with its Diophantine condition and an honest account of what “small” means; Melnikov's method stated. And the contrast with unit 01: solutions are unique and depend continuously on data, which is entirely compatible with being useless for prediction, because continuous dependence is not uniform in t.
Problems: The slides compute Hénon–Heiles at four energies, plotting the Poincaré section at each and watching the invariant curves dissolve into a chaotic sea as E passes 1/8 — the single most convincing picture in classical mechanics — then compute the largest Lyapunov exponent and confirm it positive in the chaotic region and zero on a surviving torus. The sheet: use unit 01's continuous-dependence bound to show error grows at most like e^Lt, show the rate is achieved, and conclude determinism and unpredictability are consistent rather than in tension; apply Chirikov's criterion to the standard map and compare the numerically located critical kick strength with Greene's 0.9716; and compute the 3:1 and 5:2 Jovian resonance semi-major axes against the observed Kirkwood gaps.; Checkpoint. You compute a largest Lyapunov exponent for a given map, produce a Poincaré section that constitutes evidence for chaos rather than an impression of it, and explain — quoting unit 01's theorem precisely — why a positive exponent does not contradict uniqueness.

Part C — the analytic toolkit, fields, and the crisis that ends Newtonian physics

Take a chain of oscillators to the continuum and you are holding a field, a PDE, and four more pieces of mathematics

Fourier and Sturm–Liouville theory, the Fourier transform with the complex analysis that makes it usable, asymptotics for the problems with no exact answer, and tensor fields on space. Electromagnetism is here at full strength and on purpose: it is the only place vector calculus reaches its final form, it is the historical and logical road to relativity, and F_μν is the canonical tensor field that makes the Riemann tensor's index work familiar rather than alarming. The part ends by showing, with the chain rule and nothing else, that Maxwell's equations do not survive a Galilean boost — and by finding which transformation they do survive. Eight units, 67.5 h.

From N masses to a field: the wave equation

spine

needs 07 12

Theory: The loaded string as N → ∞ with total mass fixed, giving the wave equation with c = √(T/μ) from a force balance whose small-slope approximation is stated and whose error is named. Travelling and standing waves; superposition; reflection at fixed and free ends, with and without inversion. Energy density and flux, and the local conservation law relating them. Then the string as a Lagrangian system with infinitely many degrees of freedom: the Lagrangian density and the Euler–Lagrange equation for a field, which is the first field theory in the course.
Tools: The continuum limit as a limit of second-difference operators. Classification of second-order linear PDEs by the principal symbol — hyperbolic, parabolic, elliptic — reduction to canonical form, and what each class will accept as data. Characteristics; d'Alembert's formula derived by ξ = x − ct, η = x + ct; domain of dependence and range of influence, with finite propagation speed as a theorem rather than an assumption. Hadamard well-posedness, with an explicit ill-posed problem to show the classification has teeth. Uniqueness by energy methods.
Problems: The slides pluck a string at a/L of its length: triangular initial profile, zero initial velocity, extended oddly and periodically to handle the fixed ends, then d'Alembert followed to the exact shape at t = L/4c, L/2c and 3L/4c — the flat-topped travelling kinks that high-speed photography of a real string actually shows. The sheet: a pulse meeting a fixed end and a free end by images, with both signs right; classification of u_xx + 2bu_xy + cu_yy = 0 followed by Hadamard's example, where the Cauchy problem for Laplace's equation is ill-posed with data as small as you please; and Tricomi's equation changing type across y = 0, which is what makes transonic flow hard.; Checkpoint. You solve an initial–boundary-value problem for the finite string by images, classify a given second-order PDE and state what data it will accept, and write the Lagrangian density whose Euler–Lagrange equation is the wave equation.

Fourier and Sturm–Liouville: the modes are a basis, and that is a theorem

spine

needs 18 09 10 11

Theory: Separation of variables on the finite string, and the claim that every initial shape is a superposition of its modes. Timbre: why a plucked and a struck string at the same pitch sound different, and why plucking at one seventh of the length silences the seventh harmonic — a prediction you can hear. Then the same pattern in three geometries: the string gives sines, the circular drum gives Bessel functions, the sphere gives Legendre polynomials and spherical harmonics. Heat conduction in a rod, which was Fourier's own problem. Why strings are tuneful and drums are not.
Tools: Fourier sine, cosine and complex series; orthogonality; Bessel's inequality; Parseval; completeness in L² stated with the Fejér route sketched. Dirichlet's pointwise theorem for piecewise-C¹ functions; convergence to the midpoint at a jump; uniform convergence and its failure; Gibbs' phenomenon with its 8.9490 per cent overshoot computed; the dictionary between smoothness and coefficient decay, which makes truncation error predictable; term-by-term differentiation and exactly what it costs. Then the Sturm–Liouville operator: formal self-adjointness by Lagrange's identity, the boundary conditions that make it genuinely self-adjoint, and each consequence proved — eigenvalues real and simple, eigenfunctions w-orthogonal, spectrum discrete and unbounded. Completeness via compactness of the resolvent, stated; Sturm's oscillation theorem; the Rayleigh quotient and the variational characterisation of eigenvalues, which is unit 11's calculus of variations in its second job. Bessel and Legendre recognised as two instances of one theorem; spherical harmonics from separating the Laplacian.
Problems: The slides separate the circular drum in polar coordinates, impose regularity at the centre and vanishing at the rim, and obtain frequency ratios 1 : 1.593 : 2.136 : 2.295 — irrational, therefore not a chord, which is why a drum has no pitch — then bound the fundamental above by a one-parameter trial function and compare with the exact first Bessel zero 2.4048. The sheet: compute b_n ∝ (1/n²) sin(nπa/L) for the plucked string, show b₇ vanishes at a = L/7, bound the truncation error by Parseval as N⁻³ against the struck string's N⁻¹ and say which sounds brighter; prove reality and orthogonality from Lagrange's identity, then find the boundary condition that breaks self-adjointness and see which conclusion fails first; and put Legendre's equation into Sturm–Liouville form and prove orthogonality without touching a generating function.; Checkpoint. Given a Sturm–Liouville problem you prove reality and orthogonality from self-adjointness, count the zeros of the third eigenfunction correctly, bound the lowest eigenvalue above to within a few per cent by a trial function, and expand a given initial profile with a correct statement of what kind of convergence you have.

The Fourier transform, complex analysis, and why causality is a statement about poles

spine

needs 06

Theory: The response of a linear system to any input, not only periodic ones — the susceptibility of unit 06's oscillator. Causality: a system cannot respond before it is driven, and the discovery that this innocuous statement constrains the entire frequency response. Dispersion: why glass separates colours, and why refraction and absorption are not independent. Wave packets, phase and group velocity, packet spreading.
Tools: The Fourier transform on ℝ; inversion; Plancherel; the convolution theorem; differentiation becoming multiplication; the uncertainty relation proved by Cauchy–Schwarz. Test functions and distributions, introduced properly and retroactively legitimising unit 06's Green's function. Then complex analysis: holomorphy and Cauchy–Riemann, Goursat and Cauchy's theorem, the integral formula and its consequences (analyticity, Liouville, maximum modulus, Morera), Laurent series, isolated singularities, the residue theorem, Jordan's lemma, principal values, branch points and cuts, the argument principle and Rouché, analytic continuation. Contour integration as the tool for inverting transforms, with the Bromwich contour for the Laplace transform. Asymptotics of integrals belongs here and not in unit 21: Watson's lemma, Laplace's method, stationary phase, steepest descent.
Problems: The slides invert the susceptibility by residues: both poles in the lower half-plane, close upwards for t < 0 to get exactly zero and downwards for t > 0 to collect both residues — recovering precisely the Green's function built by hand in unit 06, now with its step function explained rather than imposed. Then the proved equivalence: G(t) = 0 for t < 0 ⟺ the susceptibility is holomorphic in the upper half-plane ⟺ the Kramers–Kronig relations hold. The sheet calls in the debts: locate the complex singularity of the inverse of Kepler's equation and prove unit 09's series has radius of convergence 0.6627434…; derive the dispersion relations from analyticity and conclude that a medium which refracts must also absorb; and evaluate the Fresnel integrals with an honest arc estimate.; Checkpoint. You compute a retarded Green's function by contour integration and prove it vanishes for t < 0 by a contour-closing argument rather than by inspection, evaluate a real integral by steepest descent with every arc estimate justified, and — if you took unit 09 — settle its Laplace limit.

Perturbation theory: the series that diverges and is still right

spine

needs 06 02 08

Theory: The uncomfortable truth under most quantitative physics: the exact answer is unavailable, the approximate one is a divergent series, and truncated in the right place it is accurate to ten digits. What an asymptotic expansion means, and why convergence is neither necessary nor sufficient for usefulness. Secular terms, and why naive perturbation of an oscillator fails after one period. Boundary layers as the generic behaviour when a small parameter multiplies the highest derivative. Nondimensionalisation as the systematic way to find out which terms you are allowed to drop — unit 00's dimensional habit, now in its working form.
Tools: Order notation, asymptotic sequences, Poincaré's definition with the standard example ∑n!xⁿ; optimal truncation and the exponentially small error; Borel summation named. Regular perturbation; singular perturbation by matched asymptotic expansions, inner and outer solutions, the intermediate variable, Van Dyke's matching rule. Poincaré–Lindstedt and multiple scales for removing secular terms. WKB with its validity condition and turning points, the connection formulae stated, and the observation that this is unit 15's eikonal again. Integral asymptotics lives in unit 20 and is cited here, not repeated.
Problems: The slides run two calculations and join them. First the Duffing oscillator: naive perturbation produces a secular t sin t that destroys the approximation after t ∼ 1/ε, then Poincaré–Lindstedt gives ω = 1 + 3εA²/8 + O(ε²), checked numerically. Then the same machinery is pointed at its eventual target: u″ + u = GM/L² + εu², where the secular term is not to be removed — it is the perihelion precession, and it is real. Unit 37 completes the calculation; this is the unit that makes it honest rather than a conjuring trick. The sheet: optimal truncation of a Stieltjes series against the exact integral, confirming the error is the size of the first omitted term; a boundary layer in εy″ + y′ + y = 0 matched and compared with the exact solution at ε = 0.01; and Kolmogorov's E(k) ∝ ε^2/3k^−5/3 from dimensions alone, checked against published wind-tunnel spectra.; Checkpoint. Given a perturbed problem you nondimensionalise it, tell regular from singular, remove secular terms by Poincaré–Lindstedt or multiple scales, construct and match a boundary layer, and for a divergent asymptotic series find the optimal truncation point and state the accuracy it achieves — and you can say, for the relativistic orbit equation, why the secular term must be kept.

Continuum mechanics: the stress tensor, and conservation laws in local form

spine

needs 16 05 18

Theory: A deformable body, for which “the force on it” is not a single vector. Cauchy's stress tensor and the tetrahedron argument that produces it; pressure as the isotropic part; the symmetry of the stress tensor as angular-momentum balance. Strain, and Hooke's law reduced by isotropy to two constants. Mass and momentum conservation in local form; Euler's equations for an inviscid fluid, with Navier–Stokes stated and the viscous term identified. Sound as the linearisation of Euler's equations about rest. P- and S-waves in an elastic solid, and why only solids carry shear waves — which is how the Earth's liquid outer core was found.
Tools: The divergence theorem as the engine that converts every integral conservation statement into a local one: continuity, and momentum balance with the stress tensor supplying the flux. The material derivative and the transport theorem. Tensor fields on ℝ³ — the first use of unit 16's index machinery on a field rather than a single object. Cauchy's stress theorem proved by the tetrahedron limit: the only place in classical physics where you prove rather than assume that a physical quantity is bilinear, and the best rehearsal available for what T^μν is. Flagged explicitly: the triple (energy density, momentum density, momentum flux) assembled here becomes T^μν in unit 35, and this local conservation law is exactly the statement that its divergence vanishes.
Problems: The slides derive the speed of sound: linearise continuity and Euler about rest, close the system with p ∝ ρ^γ, and obtain c = √(γp/ρ) = 343 m/s in air at 20 °C — then show Newton's isothermal assumption gives 290 m/s at the same temperature — a factor 1/√γ, 15 per cent low — a discrepancy that stood unexplained for a century until Laplace, and a genuine historical falsification with a wrong physical hypothesis identifiable as its cause. The sheet: prove the traction is linear in the normal, hence a tensor, then prove that tensor symmetric from angular-momentum balance and identify the assumption (no intrinsic body torques) the derivation quietly uses; and compute P- and S-wave speeds from given Lamé parameters, show c_L > c_T always, explain the S-wave shadow zone, and locate an epicentre from P–S arrival differences at three stations.; Checkpoint. From an integral conservation statement you derive its local form by the divergence theorem, you prove Cauchy's theorem rather than assuming stress is bilinear, and you compute the speed of sound in air to within 2 per cent — getting the thermodynamic closure right rather than lucky.

Static fields: electrostatics and magnetostatics, the same vector calculus twice

spine

needs 05 10 19

Theory: Coulomb's law and its formal identity with Newton's, so that all of unit 05 transfers free of charge and the differences — two signs, and no shielding of gravity, so gravity accumulates while electricity screens — are the interesting part. Conductors and boundary conditions; the method of images; the multipole expansion of a charge distribution, reusing unit 10's Legendre machinery as a boundary-value tool. Magnetostatics: Biot–Savart, Ampère's law, the vector potential, gauge freedom, and the empirical fact that B has no sources, which has no electrostatic analogue. The Lorentz force, connecting back to unit 12's velocity-dependent potential.
Tools: Stokes' theorem doing real work at last: Ampère's law is the curl theorem applied to B. Poisson's and Laplace's equations; Green's functions for the Laplacian in two and three dimensions, with the dimension-dependence derived rather than asserted; uniqueness by the maximum principle; Earnshaw's theorem. Separation of variables in spherical and cylindrical coordinates using unit 19's machinery. The existence of A from div B = 0 as the Poincaré lemma, with the solenoid supplying the non-simply-connected obstruction — the same computation as unit 04's vortex, one degree up. The Helmholtz decomposition: a field decaying at infinity is determined by its divergence and its curl, which is why Maxwell's system has exactly four equations and is neither over- nor under-determined, a point almost never made explicitly.
Problems: The slides put a grounded conducting sphere in a uniform external field, solve Laplace's equation by Legendre expansion, extract the induced dipole moment and the surface charge density ∝ cos θ, then obtain the identical answer by images with two image charges and reconcile the two derivations term by term. The sheet: a point charge near a grounded plane and near a grounded sphere, then a geometry admitting no finite image set; the solenoid, with B = 0 outside but A nonzero with circulation equal to the enclosed flux, identified as unit 04's vortex one degree higher — closed but not exact; and the Helmholtz theorem stated with its decay hypotheses, proved unique, and used to predict that Maxwell's system must have four equations before you have seen them.; Checkpoint. You solve a spherical boundary-value problem by Legendre expansion and independently by images and get the same answer, produce a vector potential in a stated gauge, and state the Helmholtz theorem with its hypotheses and use it to count Maxwell's equations in advance.

Maxwell: the displacement current, and light

spine

needs 23 20 18

Theory: Faraday's law and induction. Maxwell's repair: Ampère's law is inconsistent with charge conservation for time-dependent currents, and demanding consistency forces the displacement current — a term added on purely logical grounds whose consequence is the existence of light. The complete system. Field energy and momentum; Poynting's theorem as local energy conservation; the Maxwell stress tensor, which is the first energy–momentum tensor in the course. Plane waves and polarisation. Potentials and gauge freedom; Lorenz and Coulomb gauges; retarded potentials; the Larmor formula and dipole radiation.
Tools: The wave operator and its retarded Green's function in three dimensions, computed by unit 20's contour methods rather than asserted — the iε prescription is the place where causality enters the mathematics and it is made explicit. Kirchhoff's formula and the method of descent; the strong Huygens principle, so that sharp wavefronts exist in three dimensions and not in two. Gauge transformations as redundancy of description, with the gauge-invariant fields carrying the physical content and the residual freedom in Lorenz gauge made explicit.
Problems: The slides take the divergence of Ampère's law as Ampère left it, find it contradicts charge conservation, and show exactly one term repairs it — then derive the vacuum wave equation and compute c = 1/√(ε₀μ₀) = 2.998 × 10⁸ m/s from two bench-top constants, compared with Fizeau's 1849 value of 3.13 × 10⁸, the moment light became an electromagnetic phenomenon, with an estimate of how good the measurements had to be to convince anyone in 1862. The sheet: the potentials of a point charge in Lorenz and Coulomb gauge, where the Coulomb-gauge scalar potential is instantaneous and E and B are nevertheless identical and causal; the 2D wave Green's function by descent, supported inside the light cone so a two-dimensional flash has a fading tail; and Larmor's formula, with a classical orbiting electron collapsing in 1.6 × 10⁻¹¹ s — a failure this course records and deliberately does not repair.; Checkpoint. You derive the displacement current from charge conservation alone, obtain the vacuum wave equation and c from ε₀ and μ₀, compute the retarded Green's function by contour integration with a defensible account of why that prescription is the causal one, and show two different gauges produce identical E and B.

The crisis: Maxwell's equations do not survive a Galilean boost

spine

needs 24 03

Theory: Galilean relativity, which unit 03 assumed without naming. The problem: Maxwell's equations contain c with no frame attached, and they change form under x → x − vt. Three responses and their fates — an ether, with its absurd required properties; emission theories, killed by de Sitter's binary stars; and modifying Maxwell, killed by Maxwell continuing to be right. Michelson–Morley's null result; Kennedy–Thorndike and Ives–Stilwell and what each independently pins down. Stellar aberration and Fizeau's moving water, which no ether model ever explained together. Lorentz's contraction hypothesis and local time as a patch, and Poincaré's realisation that the patches form a group.
Tools: What covariance of a PDE system under a group action means precisely. Direct computation: apply the chain rule for a Galilean boost to the wave operator and exhibit the cross terms that appear; then ask which linear substitution kills them, and derive the Lorentz transformation as the answer — before any postulate has been stated. Symmetry groups of differential operators, with the conformal group of the wave equation mentioned as the larger answer.
Problems: The slides transform the one-dimensional wave equation under a Galilean boost, display the extra terms, and show no choice of v removes them; then solve for the general linear map on (x, t) preserving the operator's form — two conditions, a one-parameter family, and the Lorentz transformation emerging with c fixed by the equation itself. Einstein's postulate arrives as a conclusion rather than an assumption, and this is the unit that lets unit 26 say “the invariant speed is c” rather than “assume the invariant speed is c”. The sheet: the expected fringe shift for the 1887 apparatus under a 30 km/s ether wind (about 0.4 fringes) against the measured bound below 0.01, with what Lorentz contraction alone would have predicted; and a demonstration that Michelson–Morley alone is consistent with several transformation laws, identifying exactly what Kennedy–Thorndike and Ives–Stilwell each add.; Checkpoint. You show by direct computation that the wave equation is not Galilean-invariant and is Lorentz-invariant, and derive the boost by demanding form-invariance — without assuming the constancy of c as a postulate.

Part D — Minkowski: relativity as the geometry of a quadratic form, and fields on it

Relativity is not a list of paradoxes; it is one quadratic form and the group that preserves it

Special relativity is presented as the structure it is: a group derived from homogeneity, isotropy and one invariant speed, acting on a vector space carrying an indefinite bilinear form. Every paradox dissolves into a statement about that form — the twin paradox is the reversed triangle inequality — and every dynamical result follows from the free action being proper time. Electromagnetism is then recognised as having been relativistic all along, field theory supplies an honest definition of the stress–energy tensor, and the part closes by taking the equivalence principle seriously and showing, from inside flat spacetime, exactly which feature of gravity Minkowski space cannot represent. Five units, 37.5 h.

Minkowski space: relativity as the geometry of a quadratic form

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needs 03 14

Theory: Einstein's two postulates and the work each does. The relativity of simultaneity as the actual content, with time dilation and length contraction as corollaries rather than postulates. The invariant interval; the twin paradox; the pole and the barn; velocity addition, relativistic Doppler, aberration and the headlight effect. Causal structure: which parts of “before” survive and which do not, so that no superluminal signalling is a theorem. A note on provenance: unit 25 earned the invariant speed's value from Maxwell, and a learner who has not done unit 25 takes it on credit here — the course says so rather than pretending otherwise.
Tools: Derive the Lorentz group from structural assumptions alone: homogeneity gives linearity, isotropy and the group axioms give a one-parameter family with a single invariant speed as the only free parameter, and unit 25's wave operator fixes it at c. Then the bilinear form η of signature (−,+,+,+); O(1,3) and its four connected components; the proper orthochronous subgroup; boosts as hyperbolic rotations with rapidity as the additive parameter, so velocity addition is nothing but the addition formula for tanh. The Alexandrov–Zeeman theorem cited: preserving causal order alone already forces the Lorentz group up to scale. Minkowski diagrams with correctly scaled axes and calibration hyperbolae, which is the only way the diagrams do not lie. The reversed triangle inequality for timelike curves, proved.
Problems: The slides do the twin paradox three ways, all agreeing: by evaluating proper time along each worldline; by counting Doppler-shifted signals, which makes the asymmetry visible as the moment the shift flips and needs no appeal to acceleration at all; and on a Minkowski diagram, tracking the travelling twin's lines of simultaneity as they jump across the turnaround. Then the general point: because the signature is Lorentzian, timelike geodesics locally maximise proper time — the opposite of the Riemannian case, and the sign that makes unit 32's geodesic principle a maximum principle. Worth being uncomfortable about for ten minutes. The sheet: prove velocity addition is tanh addition and compute the speed after ten successive 0.5c boosts; resolve the pole and barn by writing both door closings as events and transforming them; and show that a faster-than-light signal is a backwards-in-time signal in some frame, with the argument shown to fail for timelike separation.; Checkpoint. You derive the Lorentz transformation from homogeneity, isotropy, the group axioms and one invariant speed, compute proper time along any specified worldline, and prove the twin paradox is the reversed triangle inequality for timelike curves in η.

Relativistic dynamics: four-vectors, and why colliders exist

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needs 26 14 16

Theory: Four-velocity, four-acceleration, four-momentum. E = γmc², rest energy, and the case against “relativistic mass”. Invariant mass as the norm of a system's total four-momentum, which is why a box of hot gas weighs more. Conservation of four-momentum in collisions; thresholds; Compton scattering; decay; the relativistic rocket. Massless particles and why they must travel at c. Why a rigid body is impossible in relativity — Born rigidity and the Ehrenfest disc — and hence why unit 16 has no relativistic successor. Hyperbolic motion at constant proper acceleration and its Rindler horizon, which costs nothing here and is unit 30's main tool.
Tools: Four-vectors as elements of a vector space carrying a representation of the Lorentz group, with the invariant pairing giving −m²c². Proper time as arc length in Lorentzian signature; the constraint u·u = −c² and its consequence that acceleration is always orthogonal to velocity. The free-particle action S = −mc²∫dτ, whose Euler–Lagrange equations by unit 11 yield p = γmv — the first Lagrangian in the course that is not T − V and does not come from a metric on configuration space, which is the point of departure. Legendre-transforming it by unit 14 to H = √(p²c² + m²c⁴), noting how nearly degenerate the massless case is. Mandelstam invariants and the technique of squaring conservation equations to work frame-free.
Problems: The slides compute the antiproton threshold: s ≥ (4m_pc²)² evaluated in the fixed-target frame gives a beam kinetic energy of about 5.6 GeV and in the collider frame 2 × 0.94 GeV — the factor-of-three argument that justified building colliders, and the reason the Bevatron was designed at 6.2 GeV. Three lines, no frame transformation at all. The sheet: Compton's wavelength shift from four-momentum conservation in four lines; the invariant mass 2E/c² of two oppositely travelling photons, and what happens as the angle closes; and a 1/r potential with relativistic kinematics, which precesses and gives roughly one sixth of Mercury's residual — the single most instructive wrong answer in physics, because it shows time dilation alone is not gravity. Unit 37 supplies the other five sixths.; Checkpoint. You compute a reaction threshold by evaluating the invariant s in whichever frame is easiest and compare it with a published accelerator design energy, and derive p = γmv from the action −mc²∫dτ by the Euler–Lagrange equations of unit 11.

Electromagnetism was relativistic all along: the field tensor

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needs 27 25

Theory: E and B are not two fields but one field seen from a frame. The field tensor F^μν and the transformation rules; a purely magnetic field in one frame is partly electric in another. Purcell's argument: the magnetic force between a current and a moving charge is electrostatics plus length contraction, so magnetism is what relativity does to Coulomb's law at v/c ≈ 10⁻¹³, amplified by the number of charges in a wire. The covariant Maxwell equations, four collapsing to two; the Lorentz force as dp^μ/dτ = qF·u. The field's stress–energy tensor, its symmetry, tracelessness and vanishing divergence — the same statement as unit 22's continuum balance, now with energy included. Radiation reaction named, and the Abraham–Lorentz runaway solutions flagged honestly as unresolved in classical theory.
Tools: Tensor fields on Minkowski space; raising and lowering with η, the first place where unit 16's distinction between vectors and covectors is not optional. The antisymmetric (0,2)-tensor and its six components; the Levi-Civita symbol versus the Levi-Civita tensor; the dual tensor; the two invariants, one ∝ B² − E²/c² and the other ∝ E·B. The four-current and continuity as a vanishing divergence. This is where unit 16's index notation becomes fluent on a Lorentzian signature, and a reader arriving at units 32 through 37 from here finds their index work routine rather than alarming.
Problems: The slides run the wire and the test charge in full: in the lab frame a neutral current-carrying wire exerts a magnetic force on a charge moving parallel to it; boost to the charge's rest frame, where the positive and negative line densities contract differently, the wire acquires a net charge, and the force is purely electrostatic — the two forces agreeing down to the last factor of γ in the transverse-force transformation. Then the vanishing divergence of T^μν is verified to reproduce exactly Poynting's theorem in its time component and the Maxwell stress balance in its spatial ones. The sheet: given E and B in one frame, use the two invariants to decide whether a frame exists in which the field is purely electric, purely magnetic or neither, and construct the boost when it exists; and derive the Liénard generalisation of Larmor's formula from the invariant a·a, checking the non-relativistic limit.; Checkpoint. Given the field tensor and a boost you derive the E and B transformation rules and verify both invariants numerically, and you show by explicit computation that the divergence of the electromagnetic stress–energy tensor vanishes and reproduces the familiar conservation laws.

Classical field theory: Noether again, and where T^μν actually comes from

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needs 28 13 11

Theory: A field is a mechanical system whose coordinates are labelled by points of space rather than by an index, and the action is an integral of a Lagrangian density over spacetime. Three Lagrangians done explicitly: the scalar wave equation, the Klein–Gordon field, and Maxwell's, from which the source equations follow in four lines. Gauge invariance as a property of the action, forcing current conservation. Noether for fields: continuous symmetries give conserved currents and hence charges. Spacetime translation gives the canonical energy–momentum tensor — which for Maxwell comes out neither symmetric nor gauge-invariant, a genuinely surprising defect — and the Belinfante–Rosenfeld improvement repairs it. Then the definition general relativity actually uses, T^μν = (2/√−g)·δS_matter/δg_μν, automatically symmetric and gauge-invariant. This unit exists so that the right-hand side of Einstein's equation is a definition rather than an incantation.
Tools: Functional derivatives of an action over a spacetime region; the field Euler–Lagrange equations derived by unit 11's argument in four dimensions with the boundary term handled explicitly. Noether's theorem in the field-theoretic setting, proved in full. Variation with respect to a metric, introduced here in flat space with g held general, so that unit 35's Einstein–Hilbert variation is a continuation rather than a new technique. Tensor densities, √−g, and the invariant volume element.
Problems: The slides derive all of Maxwell's source equations by varying S with respect to A_μ, carefully, including the integration by parts and the boundary term — then compute T^μν twice: once as the canonical Noether current from translation invariance, obtaining the non-symmetric, non-gauge-invariant expression, and once by metric variation, obtaining the familiar symmetric one, with the Belinfante improvement term identified as exactly the difference. Two routes landing in the same place by different mathematics is the point. The sheet: from the Klein–Gordon Lagrangian derive the field equation, the energy–momentum tensor and the dispersion relation, and interpret the mass term as what makes the wave equation dispersive; and find the conserved current of the global phase symmetry of a complex scalar, then show that gauging it forces the introduction of a field with exactly the transformation law of A_μ.; Checkpoint. Given a Lagrangian density you derive the field equations, construct the Noether current of every continuous symmetry and verify its conservation on shell, and produce the symmetric energy–momentum tensor by metric variation — and for Maxwell you can state precisely how the canonical and metric-variation tensors differ and why the difference is separately conserved.

The equivalence principle, and why flat spacetime cannot hold gravity

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needs 27 05 16

Theory: Inertial and gravitational mass coincide to one part in 10¹⁵, from Eötvös through Adelberger's torsion balances to MICROSCOPE. Einstein's elevator; the weak, Einstein and strong forms stated separately because they are genuinely different claims, with an experiment that distinguishes them. Consequences derived before any geometry exists: light bends, clocks low down run slow, light climbing loses frequency. Pound–Rebka. GPS as a working system that fails without both corrections. Then the argument that ends the part: tidal forces cannot be transformed away, so there is no global inertial frame, so gravity is not a force field on Minkowski space.
Tools: Accelerated observers in Minkowski space; proper acceleration; the hyperbolic worldline from unit 27. Rindler coordinates and the metric with a position-dependent lapse — a non-constant metric on a flat spacetime, which is precisely the point that non-constant metric components are not yet gravity, and the standard inoculation against the beginner's belief that coordinate effects are physics. The Rindler horizon. Ehrenfest's rotating disc, whose spatial geometry already has circumference over diameter greater than π, so a rotating frame in flat spacetime already forces non-Euclidean spatial geometry. Unit 05's tidal tensor restated as the object no coordinate change removes — the first curvature invariant, met before curvature is defined.
Problems: The slides derive the gravitational redshift from the equivalence principle alone: in a uniformly accelerating rocket a signal takes h/c to reach the ceiling, by which time the ceiling has gained speed gh/c, so the fractional shift is gh/c². Evaluated for Harvard's 22.5 m tower that is 2.46 × 10⁻¹⁵, against Pound and Rebka's measured (2.57 ± 0.26) × 10⁻¹⁵. Then the GPS budget: +45.7 μs per day gravitational, −7.2 kinematic, net +38.5 — about 11 km of position error per day if ignored. The sheet: derive Rindler coordinates from the hyperbolic worldline, find the horizon, and show a signal from beyond it never arrives — a horizon with no gravity anywhere in sight; show a transformation to a freely falling frame kills the first derivatives of the metric but cannot kill the second, thereby proving that “gravity is not a force” and “gravity is real” are both true; and show a spin-2 field on Minkowski coupled to stress–energy must couple to its own stress–energy, stating what that self-consistency demands (Deser's argument, whose iteration produces general relativity).; Checkpoint. You derive the gravitational redshift from the equivalence principle with no geometry and match Pound–Rebka within its error bar, state the three equivalence principles with an experiment distinguishing them, and exhibit a non-constant metric on flat spacetime — thereby proving that non-constant metric components alone are not gravity.

Part E — curved space: the mathematics of “there is no global inertial frame”

The one place where three consecutive units are mathematics first, and that is the just-in-time principle rather than a lapse in it

Unit 30 ends by proving tidal forces cannot be transformed away, and the only honest response is to build the machinery that measures exactly that. Manifolds so that coordinates stop being data, connections so that differentiating a vector field means something, curvature so that the thing no coordinate change removes has a name. The payoff is immediate and precise: geodesic deviation set beside the Newtonian tidal tensor of unit 05, term for term. Four units, 37.5 h.

Manifolds: doing all of it without coordinates

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needs 16 30

Theory: Configuration space from unit 12 was a manifold all along — the double pendulum's is a torus and the rigid body's is SO(3) × ℝ³, which is why Euler angles have coordinate singularities rather than because anyone was careless. Spacetime as a manifold with no preferred coordinates, and an honest account of why general covariance, on its own, says considerably less than it sounds like.
Tools: Smooth manifolds, charts, atlases, smooth maps, diffeomorphisms. Tangent vectors defined three ways — equivalence classes of curves, derivations of germs, coordinate n-tuples with a transformation law — with the isomorphisms constructed, because reconciling them is what makes the physics literature readable. The differential of a smooth map; the cotangent space; one-forms and the dual basis, with df the canonical example. Tensor fields as sections, with unit 16's transformation law recovered as nothing more than a change of basis, so that “transforms like a tensor” becomes a theorem rather than a definition. Vector fields as derivations and as flows; the Lie bracket as the failure of flows to commute; the Lie derivative of tensors by pullback along the flow; Killing fields defined by L_Xg = 0, planted here for unit 36. Submanifolds, embeddings, Frobenius. Cited, not proved: partitions of unity, Whitney, the hairy ball theorem.
Problems: The slides take the two-sphere with two stereographic charts: write one vector field in each, verify the components transform by the Jacobian on the overlap, show no single chart covers the sphere, write the round metric in both charts and check they agree. Then the flow of ∂/∂φ, with L_∂/∂φg = 0 verified by direct computation — the first instance of the object that will make Schwarzschild's conserved energy and angular momentum appear in unit 36 — and the Lie bracket of two rotation generators recovering 𝔰𝔬(3) from unit 16, now realised by flows rather than matrices. The sheet: four candidate rules assigning four functions each to a chart of the sphere, transformed, and judged on whether they define a tensor field; an explicit computation showing L_XT is a tensor while ∂_μT^νρ is not, identifying which terms in the Lie derivative repair the transformation law; and the double pendulum's configuration space written as T² with an atlas and its kinetic energy expressed as a metric in each chart.; Checkpoint. You construct an atlas for a given manifold and verify the transition maps smooth, define a tensor field in one chart and transform it to another verifying agreement on the overlap, compute Lie brackets and Lie derivatives in coordinates, and determine whether a given vector field is Killing for a given metric.

Metric, connection, geodesics — free fall is a straight line

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needs 31 11

Theory: Free fall is geodesic motion, and the geodesic equation replaces Newton's second law. Parallel transport is what an ideal gyroscope does. The Christoffel symbols as the “gravitational field” that can be made to vanish at a point but not in a neighbourhood — which is the equivalence principle stated as a fact about connections. The Newtonian limit, recovering ẍ = −∇Φ with g₀₀ = −(1 + 2Φ/c²) and the factor of two everyone gets wrong once. Minimal coupling and the comma-goes-to-semicolon rule, with the ordering ambiguity flagged honestly and the pleasant observation that dF = 0 needs no modification because the exterior derivative never needed a connection. And the structural point that geodesic motion is not an independent postulate: a vanishing divergence of T for a localised source already implies its worldline is a geodesic.
Tools: The metric tensor and the geometry it defines: lengths, angles, volume form, raising and lowering. The problem that the partial derivative of a vector field is not tensorial. Affine connections and the covariant derivative on tensors of every type, with sign conventions fixed once; torsion; metric compatibility. The fundamental theorem of Riemannian geometry proved via the Koszul formula, so the Christoffel symbols are derived rather than posited — six lines, and the single formula the rest of the course computes with. Parallel transport along a curve as an ODE. Geodesics two ways — auto-parallel, and stationary points of length or energy by unit 11 — with the equivalence proved and the affine-parameter subtlety handled rather than waved past. Normal coordinates and the local flatness theorem: at any point one can set g to η with vanishing first derivatives but not vanishing second derivatives, which is exactly why curvature survives, and exactly the mathematical content of the equivalence principle.
Problems: The slides derive the geodesics of the round sphere twice — by the auto-parallel equation with computed Christoffel symbols, and by extremising arc length — and show they agree; then the practical trick used for the rest of the course: write the geodesic Lagrangian, take its Euler–Lagrange equations, and read the Christoffel symbols straight off, demonstrated on the Schwarzschild form before that solution has been derived. Then Newton is recovered: expand the geodesic equation to lowest order in v/c and Φ/c² and watch Newton's law appear with the factor of two fixed by requiring exactly that. The sheet: compute every Christoffel symbol of a given metric by the Lagrangian trick, check against the coordinate formula, and count the arithmetic saved; parallel-transport a vector around a spherical triangle and find the holonomy angle equals the enclosed area, with unit 33 arriving a week early; and compute the connection for ds² = dr² + r²dθ² and confirm it is polar coordinates on a plane rather than curvature.; Checkpoint. For a given metric you compute every Christoffel symbol by the geodesic-Lagrangian trick and verify against the coordinate formula, prove the Levi-Civita connection is the unique torsion-free metric-compatible one, construct normal coordinates at a point and state exactly what can and cannot be made to vanish there, and recover Newton's law from the geodesic equation with the correct factor in g₀₀.

Curvature: tidal forces, with the mask off

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needs 32 05

Theory: Geodesic deviation: two nearby freely falling particles accelerate relative to one another at a rate given by the Riemann tensor. Set this beside unit 05's Newtonian tidal equation and read off the identification of Rⁱ_0j0 with the second derivatives of the Newtonian potential — the single most important correspondence in the course, and the reason curvature is not an analogy for gravity but the same computation. Curvature as the thing no coordinate change removes, and therefore as the honest definition of a gravitational field. Why in three dimensions Ricci determines Riemann completely, and hence why there are no vacuum gravitational waves in 3D.
Tools: The Riemann tensor defined by the commutator of covariant derivatives, with the check that it is tensorial despite being built from derivatives. Its symmetries and the component count n²(n²−1)/12 — 20 in four dimensions, 6 in three, 1 in two. The first and second Bianchi identities. Sectional and Gaussian curvature; Gauss's Theorema Egregium as the statement that curvature is intrinsic. The Ricci tensor, the scalar curvature, and the Weyl tensor as the trace-free part carrying vacuum tidal effects. The Einstein tensor, and the fact that its divergence vanishes identically as a consequence of the contracted second Bianchi identity rather than by assumption — highlighted, because it is what selects Einstein's equation in unit 35.
Problems: The slides compute the Riemann tensor of the two-sphere of radius a, confirm the single independent component gives Gaussian curvature 1/a², and verify that unit 32's spherical-triangle holonomy equals the integral of curvature over it. Then geodesic deviation is derived from scratch and applied to two great circles leaving the equator a small distance apart, giving ξ″ = −ξ/a² — they converge and cross at the pole, and that is tidal force — and shown in the weak-field limit to reproduce unit 05's tidal eigenvalues (−2, 1, 1)GM/r³ term for term. The sheet: prove a 2-surface's Riemann tensor has exactly one independent component and compute it for the hyperbolic plane; prove the count n²(n²−1)/12 from the symmetries alone and explain why Ricci determines Riemann in three dimensions; and prove the Einstein tensor divergence-free from the contracted second Bianchi identity, noting this is the only reason unit 35's field equations can be consistent at all.; Checkpoint. You compute the Riemann tensor of a 2-surface and verify it has exactly one independent component, prove the contracted Bianchi identity, and show the geodesic deviation equation reduces to unit 05's Newtonian tidal equation with the correct identification of components.

Differential forms and Cartan: Stokes' theorem was always one theorem

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needs 31 24 33

Theory: Maxwell as dF = 0 and d(⋆F) ∝ the current — four equations becoming two, with the first an identity once F = dA, and gauge freedom becoming exactness. Charge conservation as the same identity applied to the other pair. Magnetic monopoles as a cohomological obstruction to writing F = dA globally. Unit 14's symplectic form as an actual object, with Liouville's theorem as the invariance of its top power ωⁿ along the Hamiltonian flow — L_Xω = 0, proved in one line by Cartan's magic formula next to the divergence proof. Unit 23's solenoid as a de Rham class.
Tools: The exterior algebra; k-forms; the wedge product; the exterior derivative characterised axiomatically with d² = 0 from Clairaut; pullbacks and the naturality of d, which is why forms are worth the trouble. Orientation, volume forms, integration over manifolds. The generalised Stokes theorem, stated once and proved on the cube, with the gradient theorem of unit 04, Green's theorem, the curl theorem of unit 23 and the divergence theorem of unit 05 recovered as its low-dimensional cases and the full dictionary to grad, div and curl written out. Closed and exact; the Poincaré lemma; de Rham cohomology, with H¹ of the punctured plane computed — closing the question unit 04 deliberately left open. The Hodge star and the metric's role in it, with the sign worked out in signature (−,+,+,+). Cartan's structure equations as a faster route to curvature, with one worked comparison against unit 33's coordinate computation.
Problems: The slides rewrite all four of Maxwell's equations in form language and expand them back into components, recovering the originals exactly including every factor and sign; then compute the de Rham class of unit 04's vortex field, show it is closed but not exact, and identify its circulation 2π as the pairing of its class with the generator of first homology — the question asked thirty units earlier, answered. The sheet: show the homogeneous Maxwell pair is the identity ddA = 0 and that charge conservation is the same identity applied to the other pair; compute H¹ of the punctured plane and of the torus and say which physical statement each dimension encodes; and recompute the curvature of one metric from unit 33 by moving frames and structure equations, counting the arithmetic saved.; Checkpoint. You express Maxwell's four equations as dF = 0 and d(⋆F) = μ₀⋆J, expand them back into components recovering the originals exactly, compute H¹ of the punctured plane — closing unit 04's open question — and reproduce one of unit 33's curvature computations by Cartan's structure equations.

Part F — Einstein: the field equations and what they predict

The endpoint is not the equation, it is the arcseconds

Two derivations of the field equations — vary the Einstein–Hilbert action, or observe with Lovelock that nothing else was ever available — then one exact solution and three numbers: Mercury's 43, the factor of two in light deflection that separates Einstein from a Newtonian corpuscle, and the Shapiro delay. Three branches extend the reach — waves, cosmology, rotating holes — and the last names precisely where this course's methods stop. Six units, 57 h.

The Einstein field equations, twice

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needs 33 30

Theory: The requirements listed before any equation is written: a tensor equation so no coordinates are preferred; second order in the metric so the initial-value problem is the familiar kind; sourced by everything carrying energy and momentum, because E = mc²; automatically compatible with local conservation since the divergence of T vanishes; and reducing to Poisson's equation in the weak, slow, static limit. The stress–energy tensor made concrete for dust, a perfect fluid and — for those who have done unit 29 — the electromagnetic field, with unit 22's continuum momentum balance recognisable as its non-relativistic shadow. The equations with the cosmological term, whose history and current status as dark energy are stated without editorialising. The vacuum equations R_μν = 0, and the crucial observation that they are not trivial because the Weyl tensor survives. Coordinate freedom and the count 10 − 4 − 4 = 2. Why global energy conservation genuinely fails in general spacetimes rather than merely looking difficult. Choquet-Bruhat's well-posed initial-value formulation cited, since it is not obvious and took forty years.
Tools: Route one, requiring unit 29: the Einstein–Hilbert action varied with respect to the inverse metric — the identity for the variation of the metric determinant, the Palatini identity, the total-derivative term that becomes the Gibbons–Hawking–York boundary contribution, and T defined as the variational derivative of the matter action, which is unit 29's definition, so the two units lock together. Route two, requiring only units 33 and 30: the contracted Bianchi identity hands you a divergence-free second-order symmetric tensor, and Lovelock's theorem says that in four dimensions the Einstein tensor plus a cosmological term is the only one — so the equation is forced rather than chosen, and the coupling is fixed by the Newtonian limit. A learner arriving without unit 29 gets route two, which is complete but does not explain where the right-hand side comes from; the unit says so plainly.
Problems: The slides do the Newtonian limit in complete detail: perturb the metric about η, take the field static and the test particles slow, use unit 32's g₀₀ result, compute the relevant Ricci component to first order, and watch the field equation become Poisson's — which is the only thing that fixes the coupling at 8πG/c⁴. Then, for those on route one, the Einstein–Hilbert action is varied from beginning to end with nothing skipped and the same constant appears. Two different routes to 8πG/c⁴ is the kind of consistency check that makes a theory believable. The sheet: show Einstein's first guess, R_μν ∝ T_μν, forces T to be trace-free by the Bianchi identity and is therefore wrong, then find the fix — reconstructing 1915 in one problem; derive the field equations from the Einstein–Hilbert action keeping every boundary term visible and identify exactly where the Gibbons–Hawking–York term is needed (this one needs unit 29); and obtain the plus and cross polarisations from the linearised vacuum equation in transverse-traceless gauge, then count independently as 10 − 4 − 4 and confirm the answers agree.; Checkpoint. You argue from the contracted Bianchi identity and Lovelock's theorem why the Einstein tensor is forced, fix the coupling constant at 8πG/c⁴ from the Newtonian limit, and linearise the equations to obtain the wave equation with the correct count of propagating degrees of freedom — and, if you have unit 29, you obtain the same field equations independently by varying the Einstein–Hilbert action.

Schwarzschild: the first exact solution, and what its two singularities are

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needs 35 13

Theory: Static, spherically symmetric, vacuum — and the Schwarzschild metric derived rather than quoted. Birkhoff's theorem: spherical symmetry forces staticity, so a pulsating star radiates nothing and the interior of a spherical shell is exactly flat — the relativistic shell theorem, and the reason there is no monopole gravitational radiation. The Schwarzschild radius: 2.95 km for the Sun, 8.9 mm for the Earth. Proper versus coordinate radial distance and time, and what an infalling and a distant observer each report. The event horizon as a one-way null surface with nothing locally remarkable about it, and the finite proper time to the singularity.
Tools: Solving the vacuum equations with the ansatz ds² = −A(r)dt² + B(r)dr² + r²dΩ² — the full computation, done once, slowly, with every nonzero Christoffel and Ricci component displayed and the integration constant fixed by unit 35's Newtonian limit. Killing vectors as infinitesimal isometries and the conserved quantities they produce along geodesics, giving E and L for free — which is unit 13's Noether theorem in geometric clothing, and worth saying out loud. Diagnosing singularities with invariants: the Kretschmann scalar 48G²M²/c⁴r⁶ is finite at the horizon and divergent only at the centre. Eddington–Finkelstein coordinates removing the horizon singularity; the Kruskal–Szekeres maximal extension and its four regions; conformal compactification and Penrose diagrams, constructed for Minkowski first so the technique is clear.
Problems: The slides derive Schwarzschild in full from the ansatz, then show the horizon is a coordinate artefact by transforming explicitly to ingoing Eddington–Finkelstein coordinates, while the centre survives every transformation because the Kretschmann scalar diverges as 1/r⁶ — and finally build the Kruskal extension step by step, tortoise coordinate to null coordinates to exponential Kruskal coordinates, with all four regions drawn. The sheet: compute the proper time for radial infall from rest to the centre and show it finite while the Schwarzschild coordinate time to reach the horizon diverges, then reconcile the two accounts without hand-waving; prove Birkhoff and state its two consequences; and compute the tidal tensor at the horizon as a function of M and find the mass above which a human survives horizon crossing.; Checkpoint. You derive Schwarzschild from the ansatz, obtain the geodesic constants of motion from its Killing vectors and name the unit-13 theorem they are an instance of, and demonstrate by an explicit coordinate transformation plus an invariant scalar that exactly one of its two apparent singularities is real.

The classical tests: three numbers, and one of them is 43

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needs 36 21 08 10

Theory: Geodesics in Schwarzschild. The effective potential with its new inverse-cube term and everything that follows: no stable circular orbit below six gravitational radii, a plunge with no Newtonian analogue, and precession of everything else. Mercury: 5600″ per century observed, 5025.6 from the precession of the equinoxes, about 531.5 from the other planets by unit 10's perturbation machinery, and 43 left over that Le Verrier attributed to a planet nobody found. Light deflection: 1.75″ at the solar limb, Eddington in 1919, and modern VLBI at 1.7510 ± 0.0004. The Shapiro delay, measured by Cassini to one part in 10⁵. Gravitational redshift, already settled in unit 30. The photon sphere at 3GM/c², which is the ring in the Event Horizon Telescope images. Modern tests named and dated: Gravity Probe B, GW150914, the S2 orbit around Sagittarius A*.
Tools: Null and timelike geodesic equations from unit 36's Killing conserved quantities. The orbit equation u″ + u = GM/L² + 3GMu²/c² — exactly unit 08's Binet equation with one extra term, which is the cleanest possible statement of what general relativity adds to Newton. Unit 21's Poincaré–Lindstedt method in its intended use: the secular term is the whole answer, giving 6πGM/c²a(1−e²) per orbit. The null case with an impact parameter giving 4GM/c²b, and the explicit comparison with the Newtonian-corpuscle calculation, which gives exactly half — so the factor of two is the discriminating prediction, and it comes from the spatial part of the metric.
Problems: The slides run Mercury end to end with a number: metric to orbit equation to perturbative solution to the answer, using a = 5.79 × 10¹⁰ m, e = 0.2056 and 415.2 orbits per century, arriving at 42.98″ per century against the observed 42.98 ± 0.04. Then light deflection both ways — 0.8756″ as a Newtonian corpuscle, 1.7512″ in general relativity — identifying precisely which metric component supplies the missing half. The sheet: find the innermost stable circular orbit at six gravitational radii from the effective potential and compute the binding energy there, 5.7 per cent of rest mass, which is the efficiency of a thin accretion disc, and compare with the observed radiative efficiency of accreting black holes; and compute the round-trip radar delay for a signal grazing the Sun, evaluate it for the 2002 superior conjunction geometry, and compare with the published Cassini measurement including its stated uncertainty.; Checkpoint. From scratch, with no formula looked up, you compute Mercury's perihelion advance to within 1″ per century of 43 and the solar light deflection as 1.75″, show the latter is exactly twice the Newtonian value and name the metric component responsible — and you can say why unit 21 forbade you from removing the secular term.

Gravitational waves

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needs 35 21 24 20

Theory: Linearised general relativity. Gauge freedom as infinitesimal diffeomorphism — structurally the same redundancy as unit 24's electromagnetic gauge, and worth meeting twice for exactly that reason. Transverse-traceless gauge; two polarisations; the effect on a ring of free test masses. The quadrupole formula, and the structural reason there is no dipole radiation where electromagnetism has one: mass conservation kills the monopole and momentum conservation kills the dipole, so gravity's lowest radiating multipole is two orders up, which is why the waves are so weak. The Hulse–Taylor binary's orbital decay matching prediction to 0.2 per cent over forty years. LIGO's GW150914: strain 10⁻²¹, three solar masses radiated in 0.2 s. The detector is a Michelson interferometer, so the same experiment that found no ether found gravitational waves.
Tools: Perturbation to first order; the trace-reversed perturbation and its sourced wave equation; the retarded solution using exactly the Green's function unit 24 computed by unit 20's contour methods. The transverse-traceless projector and an honest degree-of-freedom count: ten components, minus four gauge, minus four residual, equals two. Multipole expansion of the source in the slow-motion limit; unit 21's two-timing for the adiabatic inspiral. Why the energy carried by a wave is not a local tensor, the Isaacson average, and what that says about the phrase “energy of the gravitational field”.
Problems: The slides compute the inspiral chirp: the quadrupole moment of a circular binary, the radiated power, the balance against orbital energy giving dr/dt, the coalescence time, and ḟ ∝ f^11/3 with the chirp mass as the only combination that appears — evaluated for PSR B1913+16 to give an orbital period derivative of −2.40 × 10⁻¹² against the measured −2.423 × 10⁻¹². The sheet: start from ten components and reduce to two by fixing the gauge twice, explicit about the residual freedom at each stage; compute the power radiated by a steel bar spun as fast as its tensile strength allows and compare with LIGO's sensitivity, which is why nobody built a laboratory source; and estimate the strain from GW150914's parameters against 10⁻²¹, computing how far a 4 km arm actually moves — about 10⁻¹⁸ m.; Checkpoint. You derive the linearised wave equation, reduce ten components to two by explicit gauge fixing, exhibit both polarisations' effect on a ring of test masses, and reproduce the Hulse–Taylor orbital decay rate to within 5 per cent of the measured value.

Cosmology

branch

needs 35 22

Theory: The cosmological principle as an assumption, with the evidence for it (CMB isotropy to 10⁻⁵, galaxy surveys) stated separately from the assumption itself, and the honest note that it concerns large scales only. FLRW geometry and the three spatial curvatures. The Friedmann equations read as an energy equation with an effective potential — a direct callback to unit 08, and the cheapest honest way to understand what a cosmological constant does. Matter, radiation and Λ eras and their scalings. Cosmological redshift as the stretching of wavelengths, and why it is not a Doppler shift. The distance ladder, the Hubble parameter, and the current H₀ tension stated as an open observational problem. Big Bang nucleosynthesis as the earliest quantitative success. The horizon and flatness problems, with inflation presented with its motivation and its unresolved status.
Tools: Maximally symmetric spaces and the uniqueness of the FLRW form. Computing the Einstein tensor for FLRW — substantial, entirely mechanical, and a good test of whether unit 33's machinery is fluent. Comoving and proper coordinates; conformal time. The fluid equation obtained both from the vanishing divergence of unit 22's perfect-fluid stress tensor and as a consequence of the two Friedmann equations. Solving for single-component and mixed universes; the age integral; particle and event horizons as integrals along the light cone.
Problems: The slides derive both Friedmann equations from scratch — metric, Christoffels, Ricci, scalar curvature, then G = 8πGT with T = diag(−ρ, p, p, p) — solve the three pure cases, then integrate the flat matter-plus-Λ case with Planck's parameters (0.315, 0.685, H₀ = 67.4) to get 13.8 Gyr, locating matter–Λ equality at about z = 0.30 and noting that we live rather close to it. The sheet: integrate the Friedmann equation for the age with the Planck parameters and again with H₀ = 73, to see how much of the H₀ tension the age is sensitive to; and compute the comoving particle horizon at recombination, convert it to an angle on today's sky, find roughly one degree, and state precisely why the observed uniformity across 180° is therefore a problem.; Checkpoint. You derive the Friedmann equations from FLRW plus Einstein without consulting a reference, solve for the scale factor in the matter-, radiation- and Λ-dominated eras, and compute the age of the universe for the Planck parameters to within 0.3 Gyr of 13.8.

Rotating and charged holes, and where this course stops

branch

needs 36 15

Theory: Kerr: frame dragging, the ergosphere, the two horizons, the Penrose process and its ceiling. Reissner–Nordström. No-hair, stated. Gravitational collapse (Oppenheimer–Snyder) and the Penrose diagram of a collapse-formed hole beside the eternal one, with the regions that are not physical identified. Trapped surfaces and the Penrose–Hawking singularity theorems, stated with their energy conditions made explicit because that is precisely where they can fail. The four laws of black-hole mechanics and their thermodynamic reading; the Bekenstein–Hawking entropy and Hawking's temperature — stated as the point where this course's methods run out, since deriving them needs the quantum field theory the course has deliberately not taught.
Tools: Killing horizons and surface gravity. The expansion of null congruences and Raychaudhuri's equation, stated and interpreted. Energy conditions (null, weak, strong, dominant) and which known matter violates which. The separability of Kerr geodesics by Hamilton–Jacobi and the resulting Carter constant — the payoff of branch unit 15, and the reason that branch exists at all.
Problems: The slides locate the Kerr ergosphere, show no observer can remain static inside it, and compute the maximum energy extractable from an extremal hole by the Penrose process: 29.3 per cent of its mass. Then the Kerr Hamilton–Jacobi equation is separated and the Carter constant exhibited as the fourth constant of motion that makes the system integrable in unit 17's precise sense. The sheet: separate the Kerr Hamilton–Jacobi equation, identify the fourth constant, and say what unit 17's Liouville–Arnold picture makes of a four-constant system; and check which energy conditions a perfect fluid and a cosmological constant satisfy, locating exactly which hypothesis of the singularity theorem a positive Λ threatens.; Checkpoint. You show that inside the Kerr ergosphere no timelike worldline can hold r, θ and φ constant, compute the Penrose process ceiling, and state the singularity theorem with its energy condition and name a matter content that violates it.

Capstone — one orbit, six descriptions, and an error budget that must close

The sister course attacked one problem with four traditions and refereed them. This does the same to the two-body problem — the only system that appears in every part of this curriculum, and one with a century of published measurements to be wrong against. Estimated 20–25 h.

The brief

Describe a gravitationally bound two-body system six ways, in increasing order of what the description costs. (1) Newtonian (00, 08): closed ellipses, Kepler's laws, exact — and what that predicts for Mercury's perihelion and for Hulse–Taylor's period. (2) Analytical (11–14, and 15 if you took it): the same orbits from the action, every conserved quantity produced from a symmetry including Runge–Lenz, and the whole problem in action–angle variables so that “precession” becomes the statement that two frequencies fail to be equal. (3) Perturbative (10, 21): the roughly 531″ per century caused by the other planets, computed by osculating elements and averaging, not looked up — the step usually skipped, and the one that makes the classical test honest. (4) Special-relativistic (27): relativistic kinematics with a 1/r potential, which precesses and gives roughly one sixth of the residual; state precisely why the answer is wrong. (5) Schwarzschild geodesics (36, 37): the full answer, and the factor of two in the deflection traced to a named metric component. (6) Radiative (38, if taken): quadrupole luminosity, orbital decay, inspiral time.

Then confront the data. Assemble the arcsecond budget for Mercury and defend every line — 5025.6 from the equinoxes, about 531.5 from Venus, Jupiter and the rest, about 0.03 from solar oblateness, 42.98 from general relativity, against 5600 observed. Do the same for light. Do the same for Hulse–Taylor's period derivative against (−2.423 ± 0.001) × 10⁻¹². Integrate the Schwarzschild geodesics numerically as a cross-check on the perturbative answers, and report where the perturbation series stops being trustworthy.

The deliverable is a document in which each contribution is derived from stated principles, each approximation is named with its order, each number carries an uncertainty on both sides of the comparison, and the special-relativistic factor of six is confronted rather than skipped. Every number is computed by you; reference values appear only for comparison. If the error budget does not close to within the observational uncertainty the capstone is not finished — and you will know precisely which term you got wrong. That failing-loudly property is this course's substitute for the sister course's solver benchmark: it is the assessment, not a decoration on it.

If forty-one is too many

There is a short path, and unlike the two drafts this curriculum was synthesised from it is genuinely closed: every unit on the route has all of its needs on the route. Nothing is bought with an apology in a footnote.

The express route — 24 units, about 193 h

00 → 01 → 02 → 03 → 04 → 05 → 06 → 08 → 10 → 11 → 12 → 13 → 14 → 16 → 21 → 26 → 27 → 30 → 31 → 32 → 33 → 35 → 36 → 37

It reaches the Einstein field equations, Schwarzschild, and Mercury's 43 arcseconds in about 193 hours against the full course's 333.

What the route costs, named rather than hidden. Unit 26 derives the Lorentz group from homogeneity, isotropy and one invariant speed — which is honest mathematics — but the value of that speed is taken on credit, because unit 25 is what forces it out of Maxwell. You will believe relativity rather than having been cornered into it. Unit 35 gives you route two only: the field equations are shown to be the unique divergence-free second-order symmetric tensor by Bianchi and Lovelock, with the coupling fixed by the Newtonian limit — correct, complete, and silent on where the right-hand side comes from, which is what unit 29 answers. The stress tensor is restricted to dust, defined directly in unit 27's four-vector language; that is enough for 36 and 37 and not enough for 39.

What else you lose: the entire wave, Fourier, Sturm–Liouville, complex-analysis and electromagnetic apparatus (07, 09, 18, 19, 20, 22, 23, 24, 25, 28, 29) — and with it the historical reason anyone believed relativity, Maxwell as the canonical tensor field, Cauchy's proof that stress is bilinear, and any route into 34, 38 or 39.

A second abbreviation, for a learner who wants the mathematics more than the physics: 00, 01, 02, 04, 06, 11, 18, 19, 20, 21, 31, 32, 33 — thirteen units and about 109.5 h, ending at Riemannian geometry, which is a respectable graduate methods course in its own right with no physics prerequisite beyond unit 00. One honest caveat, since the claim matters: this one is a reading order rather than a closed subgraph. Taken literally it leaves nine needs off the list — 04 wants 03, 18 wants 07 and 12, 19 wants 09 and 10, 21 wants 08, 31 wants 16 and 30, and 33 wants 05 — so either add those or accept that the corresponding motivating physics arrives on trust.

Reading and tools

Chosen so that most units have one chapter to read rather than four books to triangulate between. Every entry says which units it is for; nothing here is required, and the slides are self-contained.

Mechanics, and where the problems come from

Landau & Lifshitz, Mechanics — 170 pages deriving all of classical mechanics from symmetry with zero padding, terse to the point of rudeness, and exactly right for a reader with maturity and no patience for drill. The primary text for Parts A and B.
Arnold, Mathematical Methods of Classical Mechanics — the same material as geometry, and the book that makes units 14, 15 and 17 feel inevitable. Read it alongside Landau, not instead.
Spivak, Physics for Mathematicians: Mechanics I — written for precisely this learner and unusually honest about where the physics literature is sloppy; slow, and worth it for units 12–14.
Goldstein, Poole & Safko — consulted when Landau is too compressed, particularly for rigid bodies.
Morin, Introduction to Classical Mechanics — every problem has a full worked solution, which is this course's format exactly, so it is the model as well as the supply.
Lim, Problems and Solutions on Mechanics and its electromagnetism and relativity companions — thousands of solved problems, the right raw material for a course whose every problem carries a solution.

The analytic toolkit, and electromagnetism

Teschl, Ordinary Differential Equations and Dynamical Systems (free from the author) — units 01, 02, 06 and 17, with Picard–Lindelöf at the right level of proof.
Strogatz, Nonlinear Dynamics and Chaos — unit 02's pictures, and the best-written book on this list.
Hubbard & Hubbard, Vector Calculus, Linear Algebra, and Differential Forms — units 04, 05 and 34 in one volume, rigorous without austerity; Spivak's Calculus on Manifolds is the 140-page alternative and the proof of Stokes' theorem to own.
Gelfand & Fomin, Calculus of Variations — unit 11: short, Dover-cheap, and including the conjugate-point theory most books omit.
Strauss, Partial Differential Equations — unit 18, the cleanest treatment of classification and well-posedness at this level; Evans chapters 2–4 when you want more.
Stein & Shakarchi, Fourier Analysis and Complex Analysis — units 19 and 20, two volumes of one series, which keeps notation consistent across the seam where the course needs both at once.
Bender & Orszag, Advanced Mathematical Methods for Scientists and Engineers — unit 21, the only book that takes divergent series seriously and the source of several of its problems.
Griffiths, Introduction to Electrodynamics — units 23 and 24, the standard for good reason; Zangwill when Griffiths is too brief, Jackson consulted and never read.
Purcell & Morin, Electricity and Magnetism — the source of unit 28's wire-and-charge argument in its original form; Schwartz, Principles of Electrodynamics (Dover) derives magnetism from electrostatics plus a boost, which is unit 28's worked example written as a whole book.

Relativity, geometry, and Part F

Taylor & Wheeler, Spacetime Physics (free PDF from the authors) — unit 26's geometric framing and the source of “proper time is arc length”.
Rindler, Relativity: Special, General and Cosmological — the group-theoretic derivation done carefully, and the best account of Born rigidity for unit 27.
Lee, Introduction to Smooth Manifolds (2nd ed., chapters 1–17 and 19) — unit 31's reference, and the book that makes “a tensor is a multilinear map” stick; Introduction to Riemannian Manifolds for unit 32, with the Levi-Civita uniqueness proof done properly, do Carmo the shorter alternative.
O'Neill, Semi-Riemannian Geometry with Applications to Relativity — the one geometry book written in indefinite signature from the start, and therefore the one that never has to apologise later; the ideal bridge from unit 33 into Part F.
Frankel, The Geometry of Physics — spans units 04, 16, 31, 32 and 34 in one volume and connects them to the physics as it goes.
Carroll, Spacetime and Geometry (lecture-note version free) — the main text for Part F, best on physical motivation and on getting to Schwarzschild quickly, and matching this course's (−,+,+,+) conventions.
Wald, General Relativity — the rigorous account and the obvious second book for someone who writes proofs; chapters 1–6 track units 31–37 almost exactly, and 9–12 are where units 36 and 40 get their precision.
Misner, Thorne & Wheeler, Gravitation — consulted rather than read, but unit 33's geodesic-deviation-versus-tidal-tensor comparison is theirs and worth reading in the original.
Hartle, Gravity — physics-first, useful as a second pass over unit 37's tests. Zee, Einstein Gravity in a Nutshell — unit 35's Einstein–Hilbert variation, done more readably than anyone. Poisson, A Relativist's Toolkit — unit 35's boundary term and unit 40's horizons and surface gravity.
Maggiore, Gravitational Waves, Volume 1 — unit 38. Weinberg, Gravitation and Cosmology — unit 39, and worth an afternoon as the dissenting view that deliberately downplays geometry.
Lightman, Press, Price & Teukolsky, Problem Book in Relativity and Gravitation — 475 problems with complete worked solutions. For units 26–40 this is not a supplement; it is the problem sets.

What this proposal still needs from you

Eight open questions, two of which — the format of the solutions, and how much electromagnetism — should be settled before unit 00 is built rather than after. None of the rest blocks starting.

Open

How much electromagnetism, and this is the biggest sizing decision left. Three spine units before relativity (23, 24, 25) plus two after (28, 29), 37.5 h in all, on four grounds: it is the only place vector calculus reaches its final form, since Stokes' theorem does no real work until Ampère's law; it makes the invariance of c a conclusion rather than a decree; F_μν is the canonical tensor field that makes unit 33's index work routine; and its stress–energy tensor is a standard source. Electrostatics and magnetostatics are merged into one unit because unit 19 already gave you Legendre, Bessel and Green's functions. The alternative is one unit that states Maxwell and moves on, saving about 30 h and turning relativity into a postulate you accepted. Your call changes 23–25, 28, 29 and 38.
Should differential forms (34) be spine rather than branch? One draft made forms the first treatment of vector calculus, at unit 05. That was rejected: teaching flux as a 2-form to someone who has never computed a flux produces a learner who can prove d² = 0 and cannot guess which way a field points. But the reverse case is real — Maxwell as dF = 0, gauge freedom as exactness, the solenoid as a de Rham class, and Cartan making every curvature computation about four times shorter. If you take it, take it immediately after unit 33 rather than at the end, and accept a translation tax on every textbook page.
Derived or postulated field equations? Unit 35 offers both routes and unit 29 is spine so that the full course does the derivation. Route two is complete, forced, and silent about where T^μν comes from; route one costs unit 29 and answers exactly the question a mathematician asks first. Both are kept, and 35 needs only route two's prerequisites — which is what lets the express route be genuinely closed. If you never intend to take the express route, 29 could become a formal prerequisite of 35 and the unit would be cleaner.
Signature convention. (−,+,+,+) throughout, matching Carroll, Wald, MTW, O'Neill and Weinberg; the particle-physics literature uses (+,−,−,−), and unit 27's Mandelstam invariants are conventionally written there. Whichever you choose must be chosen once and printed on every slide: a sign error at unit 32 propagates silently to unit 37 and surfaces as a wrong number for Mercury, which is the worst possible place to discover it.
Does any of this get formalised? You write Lean and Rocq, and four things here are tractable: Picard–Lindelöf (01, mathlib already has uniqueness), the fundamental lemma of the calculus of variations (11), the uniqueness of the Levi-Civita connection (32), and the Riemann tensor's symmetries and component count (33). Formalising any of them would give this course what the sister course's solver comparisons gave it — an external referee that does not care how convincing the argument felt. It is also a substantial detour. Worth deciding before unit 01 is built.
Where the worked solutions live. Every problem needs a full solution plus a hint ladder, which is a large amount of LaTeX-heavy prose. The sister course settled on one Org file per unit with #+startup: content, after discovering that HTML <details> folds only in a browser. The same shape would work, but heavy mathematics renders badly in Org's LaTeX fragments. The alternatives are a compiled PDF per unit (typesets best, folds worst) and a second reveal.js deck revealed one rung at a time (native hint ladder, worst to write). Settle this before unit 00: retrofitting forty units of mathematics into a different format is exactly the work nobody does.
The size, and what to cut first. If 333 h is too long, cut in this order: unit 17 (chaos), which nothing in the course depends on at all and which is a genuine detour from the Newton-to-Einstein line; then unit 40, which depends on branch 15, so that a chain of two branches is a lot of machinery for one payoff; then unit 09, accepting that unit 19's drum must take Bessel on trust and unit 20 loses its best debt to pay off.
Judged and deliberately not fixed. Unit 02 feeds almost nothing downstream and stays on the spine as a pedagogical bet you may disagree with. Unit 19 merges Fourier series and Sturm–Liouville into a 10.5 h unit, the largest size here and shared with units 20, 33, 36 and 37; splitting it is defensible and would make the count 42. Units 15 and 17 exist partly for pleasure, and this says so rather than dressing them up as prerequisites. A quantum coda would fit naturally after unit 21 and is excluded only because you did not ask.

Deliberately excluded

Quantum mechanics, in every form. The largest exclusion and the most tempting to breach: by unit 19 you own Sturm–Liouville theory and L², by unit 15 Hamilton–Jacobi and the eikonal, by unit 21 WKB — the Schrödinger equation is about two hours away at three separate points. Out because you named relativity as the destination and because quantum mechanics is a second course of comparable size. The seams are flagged in one sentence each and not pursued.
Thermodynamics and statistical mechanics, except the single adiabatic relation unit 22 needs for the speed of sound. Unit 14's Liouville theorem is where statistical mechanics would naturally begin, and the slide says so and stops. The cost is real: unit 40's black-hole thermodynamics has to be stated rather than understood.
Quantum field theory and everything downstream. Unit 29 deliberately builds classical field theory in the shape QFT later uses, and that preparation is the whole of the contribution. Unruh is gestured at by unit 30's Rindler horizon and Hawking radiation stated in unit 40, neither derived.
Laboratory work and error analysis as disciplines, per your request — with one qualification. Published measurements appear constantly (Pound–Rebka, Cassini, Hulse–Taylor, LIGO, Planck, MICROSCOPE, VLBI) with their uncertainties, because a prediction compared against a number with no error bar is not a test. You are asked to be right about nature, not to measure it.
Fluid dynamics past the linear-acoustic level: turbulence, vorticity dynamics, aerofoils, the whole of aerodynamics. Unit 22 takes the stress tensor and the sound wave and stops. What fluids teaches best about dimensional analysis — Reynolds number, Kolmogorov scaling, the boundary layer as a singular perturbation — is harvested into units 00 and 21 as problems.
Optics as a subject. Fermat (11), the eikonal (15, 21) and dispersion (20) are in because they are load-bearing. Lenses, interferometry, diffraction, coherence and imaging are out: a beautiful application of unit 19's Fourier analysis, and on no path to Einstein's equations.
Circuits, AC analysis, media, transmission lines, antennas, waveguides and plasma physics — the applied two-thirds of a standard electromagnetism course. Units 23–25 take Maxwell as far as radiation and then turn towards relativity, which is the only reason electromagnetism is here.
Spinors, the Dirac equation and SL(2,ℂ) → SO⁺(1,3). A genuine gap, and the exclusion this proposal is least comfortable with, since spinors are arguably the deepest thing special relativity contains. Out because nothing on the road to the Schwarzschild tests requires them and adding them honestly costs two units.
Gauge theory beyond the U(1) example in unit 29's second problem: fibre bundles, connections on principal bundles, the tetrad formalism. Unit 34 goes as far as Cartan's structure equations and stops.
Numerical relativity, the ADM 3+1 split, and the Cauchy problem for Einstein's equations. Named in units 35 and 38, taught in neither — and the subject that actually produces the waveform templates unit 38 compares against.
Alternative theories of gravity: Brans–Dicke, f(R), TeVeS. Unit 35's Lovelock argument is the right lens on the whole family, and it is one slide rather than a unit.
Measure-theoretic rigour and functional analysis proper. L² is used as an inner product space, completeness of eigenfunction systems is cited, Lebesgue integration is named where it matters and never used, distributions are introduced operationally in unit 20. You could prove these and would enjoy it; it would cost three units that buy no physics.
Numerical methods as a discipline — discretisation, stability, stiffness, symplectic integrators, convergence order. Several problems end in arithmetic; none end in a library.
History of science narrated for its own sake. Dates, names and corrections to the received story appear throughout, but only where the history is the argument: Newton's isothermal sound speed, Maxwell's displacement current, Michelson–Morley, Le Verrier's missing planet.