Apostol — Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability (Fast Track 0.3) — Audit + Gap Plan

Book: Tom M. Apostol, Calculus, Volume II: Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability (2nd edition, John Wiley & Sons, 1969). ISBN 978-0-471-00007-5.

Fast Track entry: 0.3 (the §0 prerequisites slot — Apostol Vol 2 is the multi-variable calculus + linear-algebra-introduction prerequisite for the whole Codex spine; it sits alongside Apostol Vol 1 (0.2) and Lang Basic Mathematics (0.1) as the foundation that every subsequent strand assumes operationally).

Purpose: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite per the orchestration protocol). Punch-list of new units + deepenings to reach the equivalence threshold (docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4). Not a full P1 audit (no line-number-level theorem / exercise inventory; Apostol Vol 2 has roughly 800 pages, ~400 named results, and ~2400 exercises across 16 chapters — line-level inventory deferred).

The audit surface is the largest of any single Tier-α book: Codex 01-foundations / linear-algebra ships only 4 templated units (field, vector-space, bilinear-quadratic-form, plus group in 01.02), and 02-analysis ships only topology + functional-analysis chapters with no multi-variable calculus, ODE, or vector calculus content whatsoever. Vector-calculus-on-manifolds pieces (Stokes, integration on manifolds, differential forms) are well covered in 03.04.*, so the integration side has a [DEEPEN] / [ENRICH] route there. Most of the rest is genuinely [NEW].

§1 What Apostol Vol 2's book is for

Apostol Vol 2 is the canonical American honours-calculus second-year textbook: paired with Vol 1 it forms the rigorous calculus + linear algebra foundation that two generations of MIT, Caltech, and elite- university science / engineering / mathematics undergraduates were trained on. It is unusual among multi-variable calculus textbooks because it builds linear algebra first (Chapters 1-7) and then uses it to do multi-variable calculus properly (Chapters 8-13), with applications to differential equations and probability afterwards. Where competing texts (Marsden-Tromba, Hubbard-Hubbard, Edwards) develop linear algebra as needed for vector calculus, Apostol develops it as a self-contained abstract-algebra course in its own right and then uses it. This is why the book is the foundation for the whole Codex spine: a reader who finishes Vol 2 has the linear-algebra vocabulary required by every subsequent strand (manifolds, bundles, representation theory, spectral theory, linear PDE, symplectic geometry, etc.).

The book is organised as 16 chapters across roughly five Parts:

Part 1 — Linear Analysis (Chapters 1-5): the linear-algebra spine. Linear spaces, linear transformations and matrices, determinants, eigenvalues and eigenvectors, eigenvalues of operators on Euclidean spaces (inner-product theory: orthogonality, the spectral theorem for symmetric / Hermitian operators, the principal-axes theorem for quadratic forms, the singular-value-decomposition-adjacent material).
Part 2 — Linear Differential Equations (Chapters 6-7): applications of the linear-algebra spine. Chapter 6 develops linear ODEs with constant coefficients (homogeneous and inhomogeneous, the characteristic polynomial), method of variation of parameters, the Wronskian; Chapter 7 generalises to systems of linear differential equations using the exponential-of-matrix machinery.
Part 3 — Multi-variable Differential Calculus (Chapters 8-9): partial derivatives, the gradient, the Jacobian matrix, the chain rule, the Taylor formula in several variables, sufficient conditions for extrema; applications to geometry (level sets, tangent planes, normal vectors, extrema with constraints, Lagrange multipliers, the implicit-function theorem).
Part 4 — Multi-variable Integral Calculus and Vector Calculus (Chapters 10-12): line integrals, multiple integrals (Riemann integrability, Fubini, change of variables / Jacobian formula), surface integrals; the integral theorems (Green's theorem in the plane, Stokes's theorem on surfaces, the divergence theorem in $R^{3}$ ).
Part 5 — Probability and Numerical Analysis (Chapters 13-15): set functions and elementary probability (Chapters 13-14: $σ$ -algebras- in-disguise, probability as a set function, conditional probability, random variables in discrete and continuous form, expectation, variance, standard distributions, weak law of large numbers, Chebyshev's inequality); numerical analysis (Chapter 16: polynomial approximation, Lagrange interpolation, finite differences, numerical integration, Newton's method, fixed-point iteration).

Distinctive Apostol-Vol-2 choices:

Linear algebra before multi-variable calculus. The order matters: linear maps, matrices, determinants, eigenvalues — first — then the differential of $f : R^{n} \to R^{m}$ is defined as the linear map $d f_{p}$ , not as the matrix of partials. Apostol teaches the modern (Cartan-Dieudonné) framing in 1969 — early.
Linear-spaces-first treatment. Chapter 1 defines a real or complex vector space abstractly (axioms), develops basis / dimension / subspace / quotient, then specialises to $R^{n}$ . Distinguishes sharply from Strang-style "computations on $R^{n}$ " approach.
Determinants axiomatically. Chapter 3 develops the determinant from three axioms (multilinear, alternating, $det I = 1$ ) before any expansion-by-minors formula. Modern.
Spectral theorem on inner-product spaces. Chapter 5 proves the spectral theorem for symmetric / Hermitian operators on finite-dimensional inner-product spaces, including the principal-axes theorem for quadratic forms and the simultaneous-diagonalisation result. Done before any reference to functional analysis.
ODE via linear algebra. Chapters 6-7 do not present ODEs as a recipe-based subject; they treat the solution space of an $n$ -th-order linear ODE as a vector space and the characteristic polynomial as the eigenvalue problem of the differentiation operator. Apostol's framing is the modern one Arnold popularised later.
Differential as a linear map. The differential of $f$ at $p$ is defined intrinsically as the linear map $df_p : \mathbb{R}^n \to \mathbb{R}^m $s u c h t ha t$ f(p+h) = f(p) + df_p(h) + o(|h|)$. The Jacobian matrix is then the matrix of $d f_{p}$ in standard bases. Critical for Codex's eventual development of derivatives on manifolds.
Implicit and inverse function theorems with full proofs. Chapter 9 proves both rigorously (contraction mapping argument, in finite-dimensional form). These are the two theorems every working mathematician uses constantly and Apostol's proofs are clean and modern.
Lagrange multipliers with full proof. Not heuristic; full proof via the implicit function theorem.
Vector calculus integral theorems. Green, Stokes (classical surface form), divergence — proved (in modern $C^{1}$ generality, not just $C^{\infty}$ ). Apostol's Stokes is the classical form, not yet the differential-forms / manifold form (which Codex 03.04.05 already covers).
Probability axiomatically. Chapter 13 introduces probability via Kolmogorov axioms (probability as a $σ$ -additive set function on a $σ$ -algebra), then specialises to discrete and continuous cases. Modern; rare in 1969 calculus textbooks.
Numerical analysis appendix. Chapter 16 introduces Lagrange interpolation, finite differences, Newton-Cotes integration, Newton's method, fixed-point iteration. Codex has no numerical analysis content — a deliberate scope choice; this chapter is largely out of scope for the FT spine.
Worked-example density. Apostol's 800 pages contain ~150 fully worked examples and ~2400 exercises. The exercise layer is the largest single layer-2 gap on the Codex side.

Pedagogical position in the Codex curriculum. Apostol Vol 2 is the foundation book — every subsequent Tier-α / β book assumes the reader has the linear-algebra and multi-variable-calculus operational competence the book delivers. Closing the Apostol Vol 2 equivalence gap is prerequisite work for the whole Fast Track campaign. It produces the 01-linear-algebra and 02-analysis content that Hatcher / Bott-Tu / Lawson-Michelsohn / Sternberg / Cannas / Donaldson all silently assume.

Apostol Vol 2 ends before measure theory proper (Chapter 13 stays in elementary probability), before complex analysis (covered in Vol 1 appendix, not here), and before nonlinear ODEs (Arnold's ODEs picks that up). Operational, not foundational on those three.

§2 Coverage table (Codex vs Apostol Vol 2)

Cross-referenced against the Codex content tree (find content -name "*.md" | sort). ✓ = covered at Apostol-equivalent depth, △ = topic present but Codex unit shallower (typically templated v0.5 prose, or covered tangentially in a different framing), ✗ = not covered.

The status convention in the rightmost-column note matches peer audits (Hatcher / Donaldson / Cannas).

Part 1 — Linear Algebra (Chapters 1-5)

This is the largest single coverage gap. Codex 01.01-linear-algebra/ ships only 4 units (field, vector-space, bilinear-quadratic-form). The operational vocabulary of linear algebra — matrix, linear map, kernel, image, rank, basis, dimension, determinant, eigenvalue, diagonalisation, inner product, orthogonality, spectral theorem — is essentially absent as named units, though it is silently used everywhere downstream (in tangent spaces, bundle structure groups, representation theory, symplectic linear algebra).

Apostol topic	Codex unit(s)	Status	Note
Real / complex vector space (axioms)	`01.01.03-vector-space`	△	Templated unit shipped; covers the axioms. Apostol Ch. 1 reference.
Subspace, span, linear independence, basis	`01.01.03-vector-space` (mentioned)	△	Mentioned inside the vector-space unit; not its own unit. Gap (medium priority — should be its own unit).
Dimension theorem, $dim (U + V) + dim (U \cap V) = dim U + dim V$	—	✗	Gap. Apostol Ch. 1 §1.13.
Linear transformation (definition, kernel, image)	—	✗	Gap (high priority). Apostol Ch. 2. No standalone Codex unit on linear maps — silently assumed in `03.02.01-smooth-manifold` (tangent map), `03.05.04-vector-bundle-connection` (parallel transport), and throughout `07.*`.
Rank-nullity theorem	—	✗	Gap (high priority). Apostol Ch. 2 §2.7. Foundational.
Matrix of a linear map (in fixed bases); change of basis	—	✗	Gap (high priority). Apostol Ch. 2 §2.13-2.16.
Matrix multiplication, inverse, transpose, similarity	—	✗	Gap (high priority). Apostol Ch. 2 §2.17-2.21. Operational vocabulary used everywhere downstream.
Determinant (axiomatic + expansion + properties)	—	✗	Gap (high priority). Apostol Ch. 3. The axiomatic treatment (multilinear, alternating, $det I = 1$ ) is the modern standard; Codex assumes it silently in `03.04.02-differential-forms` (top form on $R^{n}$ ) without ever introducing it.
Eigenvalue, eigenvector, characteristic polynomial	—	✗	Gap (high priority). Apostol Ch. 4. Used silently in `02.11.03-unbounded-self-adjoint`, `03.05.09-curvature` (sectional curvature is an eigenvalue computation), and throughout `07-representation-theory`.
Diagonalisability; condition for diagonalisability	—	✗	Gap. Apostol Ch. 4 §4.10-4.13.
Cayley-Hamilton theorem	—	✗	Gap (medium). Apostol Ch. 4 §4.16.
Minimal polynomial, generalised eigenspaces, Jordan canonical form	—	✗	Gap (medium). Apostol Ch. 4 §4.20 + Ch. 7 (used in matrix exponentials). Note: Apostol gives a partial Jordan-form treatment; Hoffman-Kunze fills it.
Inner product space, Gram-Schmidt, orthogonal projection	`02.11.07-inner-product-space`	△	Codex unit covers the infinite-dimensional / Hilbert framing; Apostol's finite-dimensional treatment + Gram-Schmidt + projection is implicit but not dedicated.
Orthogonal / unitary operators, $O (n)$ / $U (n)$ as automorphism groups	`03.03.03-orthogonal-group`	△	Group-theoretic / Lie-group framing of $O (n)$ is in `03.03.03`; the linear-algebraic treatment (orthogonal operator preserves inner product) is not explicit.
Spectral theorem (symmetric / Hermitian operators, finite-dim)	`02.11.03-unbounded-self-adjoint`	△	Codex covers the unbounded / functional-analysis version; Apostol's clean finite-dimensional spectral theorem is missing. High priority — load-bearing.
Principal-axes theorem for quadratic forms	`01.01.15-bilinear-quadratic-form`	△	Templated unit on bilinear / quadratic forms; the principal-axes / diagonalisation result is in scope but not detailed.
Bilinear and quadratic forms, signature, Sylvester's law of inertia	`01.01.15-bilinear-quadratic-form`	△	Templated; Sylvester's law not stated cleanly. Medium priority deepening.

Part 2 — Linear Differential Equations (Chapters 6-7)

Codex has zero ODE coverage anywhere. This is a strand-level gap: the Arnold Mathematical Methods of Classical Mechanics audit (plans/fasttrack/arnold-mathematical-methods.md) treats ODE infrastructure as deferred to a future Arnold ODEs / Apostol Vol 2 batch; Apostol Vol 2 is the place where it gets created.

Apostol topic	Codex unit(s)	Status	Note
First-order linear ODE, integrating factor	—	✗	Gap (high priority). Apostol Ch. 6 §6.7.
Existence / uniqueness for linear ODEs	—	✗	Gap. Apostol Ch. 6 §6.6. (General Picard-Lindelöf is in Arnold ODEs; Apostol covers the linear case rigorously.)
$n$ -th order linear ODE with constant coefficients (homogeneous)	—	✗	Gap (high priority). Apostol Ch. 6 §6.9-6.12. The characteristic polynomial as the eigenvalue problem of $D = d / d x$ .
Inhomogeneous linear ODE; method of undetermined coefficients; variation of parameters	—	✗	Gap. Apostol Ch. 6 §6.13-6.18.
Wronskian, linear independence of solutions	—	✗	Gap. Apostol Ch. 6 §6.5.
Systems of linear ODEs $\overset{x}{˙} = A x$	—	✗	Gap (high priority). Apostol Ch. 7.
Matrix exponential $e^{t A}$ , computation via Jordan form	—	✗	Gap (high priority). Apostol Ch. 7 §7.7-7.10. Used silently in `03.03.01-lie-group` (one-parameter subgroups).
Phase portraits, qualitative theory of $\overset{x}{˙} = A x$	—	✗	Gap (medium). Apostol Ch. 7 §7.13-7.15. Bridges to Arnold dynamical-systems framing.

Part 3 — Multi-variable Differential Calculus (Chapters 8-9)

Codex has zero multi-variable calculus chapter. This is the second-largest gap. Pieces appear scattered (03.02.01-smooth-manifold mentions tangent maps, 03.04.04-exterior-derivative uses partial derivatives) but none of the foundational multi-variable calculus is shipped as named units.

Apostol topic	Codex unit(s)	Status	Note
Limits, continuity for $f : R^{n} \to R^{m}$	—	✗	Gap. Apostol Ch. 8 §8.2-8.4.
Partial derivative; existence vs. continuity	—	✗	Gap (high priority). Apostol Ch. 8 §8.5-8.6. Foundational.
Differentiability (linear-map definition); the differential $d f_{p}$	—	✗	Gap (high priority). Apostol Ch. 8 §8.10-8.13. The modern / Cartan-Dieudonné definition.
Gradient $\nabla f$ , directional derivative	—	✗	Gap. Apostol Ch. 8 §8.14.
Jacobian matrix; chain rule (matrix form)	—	✗	Gap (high priority). Apostol Ch. 8 §8.18-8.21.
Mean-value theorem, Taylor's theorem in several variables	—	✗	Gap. Apostol Ch. 9 §9.4-9.7.
Sufficient conditions for extrema; Hessian	—	✗	Gap. Apostol Ch. 9 §9.8-9.9.
Implicit function theorem	—	✗	Gap (high priority). Apostol Ch. 9 §9.10-9.14. Used silently in `03.02.01-smooth-manifold` (regular-value theorem follows from IFT).
Inverse function theorem	—	✗	Gap (high priority). Apostol Ch. 9 §9.16. Same — silently assumed throughout `03.*`.
Lagrange multipliers, extrema with constraints	—	✗	Gap. Apostol Ch. 9 §9.18-9.22.
Tangent plane to a level surface; normal vector	—	✗	Gap. Apostol Ch. 9 §9.23. Bridges to `03.02.01-smooth-manifold` (tangent space generalises this).

Part 4 — Multi-variable Integral Calculus (Chapters 10-12)

This is where the Codex 03-modern-geometry chapter has substantial overlap. The differential-forms / Stokes machinery in 03.04.* covers the modern abstract version of these theorems; what's missing is the classical $R^{n}$ -flavoured presentation (Riemann integral, Fubini, classical Green/Stokes/divergence in components).

Apostol topic	Codex unit(s)	Status	Note
Line integral $\int_{γ} P d x + Q d y$	—	✗	Gap. Apostol Ch. 10 §10.4-10.10. The classical-form line integral — Codex `03.04.02-differential-forms` has the abstract 1-form integration but not the classical $P d x + Q d y$ presentation.
Path-independence, conservative fields, scalar potentials	—	✗	Gap. Apostol Ch. 10 §10.14-10.18. The proper-1-form / closed-vs-exact discussion; in modern language, the de Rham cohomology question $H_{d R}^{1}$ . Codex `03.04.06-de-rham-cohomology` has the abstract version.
Riemann integral over rectangles in $R^{n}$	—	✗	Gap (medium priority). Apostol Ch. 11 §11.2-11.6. Operational definition of the multiple integral.
Fubini's theorem (Riemann form)	—	✗	Gap (high priority). Apostol Ch. 11 §11.7. The classical Fubini for continuous integrands; the measure-theoretic Fubini comes later.
Change of variables / Jacobian formula for multiple integrals	—	✗	Gap (high priority). Apostol Ch. 11 §11.26-11.30. The classical $\int_\Omega f = \int_{\Omega'} (f \circ g)
Polar / cylindrical / spherical coordinates as worked examples	—	✗	Gap. Apostol Ch. 11 §11.32-11.34.
Surface integral $\iint_{S} f d S$ , $\iint_{S} F \cdot d S$	`03.04.03-integration-on-manifolds`	△	Codex has the abstract / manifold-form integration; the classical parametric-surface presentation is not shipped.
Green's theorem in the plane	`03.04.05-stokes-theorem` (limit case)	△	Codex has Stokes in differential-forms generality; Green's theorem is the $n = 2$ classical case. Worth a dedicated treatment. [ENRICH] in `03.04.05` rather than full new unit.
Stokes's theorem (classical, surface form)	`03.04.05-stokes-theorem`	△	Codex covers the modern manifold-form Stokes; the classical $\int_{\partial S} F \cdot d r = \iint_{S} (\nabla \times F) \cdot d S$ presentation is the Apostol form. [ENRICH] in `03.04.05`.
Divergence theorem in $R^{3}$	`03.04.05-stokes-theorem` (limit case)	△	Codex has the manifold version; classical divergence theorem is the $n = 3$ case with $\nabla \cdot F$ . [ENRICH].
Vector fields, divergence, curl, gradient — operational identities	—	✗	Gap (medium priority). The classical-vector-calculus identities ( $\nabla \times \nabla f = 0$ , $\nabla \cdot \nabla \times F = 0$ , vector-triple-product identities). Codex has them implicitly via $d^{2} = 0$ . Worth a dedicated bridge unit translating between classical $\nabla$ -operator and modern $d$ on forms.

Part 5 — Probability and Numerical Analysis (Chapters 13-15, 16)

Apostol topic	Codex unit(s)	Status	Note
$σ$ -algebra, probability as a set function (Kolmogorov axioms)	—	✗	Out of scope (P4). Apostol Ch. 13. Pre-measure-theoretic probability. Defer until measure-theory chapter exists in Codex.
Conditional probability, independence, Bayes	—	✗	Out of scope (P4). Apostol Ch. 13.
Random variable, distribution function, expectation, variance	—	✗	Out of scope (P4). Apostol Ch. 14.
Discrete distributions (binomial, Poisson, geometric)	—	✗	Out of scope (P4). Apostol Ch. 14 §14.10.
Continuous distributions (normal, exponential, uniform)	—	✗	Out of scope (P4). Apostol Ch. 14 §14.20.
Chebyshev's inequality, weak law of large numbers	—	✗	Out of scope (P4). Apostol Ch. 14 §14.30.
Lagrange interpolation, finite differences	—	✗	Out of scope (P4). Apostol Ch. 16. Codex has no numerical-analysis content; deliberate scope choice.
Newton-Cotes numerical integration	—	✗	Out of scope (P4). Apostol Ch. 16.
Newton's method, fixed-point iteration	—	✗	Out of scope (P4). Apostol Ch. 16.

Aggregate coverage estimate

Theorem layer: ~15% of Apostol Vol 2's named theorems map to Codex units. Concentrated almost entirely on the vector-calculus end (Stokes / Green / divergence in 03.04.05, integration in 03.04.03) and a small slice of linear algebra (01.01.03, 01.01.15, 02.11.03, 02.11.07). After priority-1: ~55%. After priority-1+2: ~80%. After priority-3 (out-of-scope cuts honoured): ~85% modulo deliberate-out-of-scope probability and numerical analysis.
Exercise layer: ~1%. Apostol's ~2400 exercises vs. Codex's effectively-zero coverage on the linear-algebra and multi-variable side. Closing this requires a dedicated exercise-pack pass after the theorem-layer batch lands — likely the largest single exercise-pack production on the Fast Track.
Worked-example layer: ~5%. Apostol's ~150 worked examples cover matrix computations, Jacobian computations, line-integral computations, ODE solutions, eigenvalue computations — almost none of which appear as Codex worked examples.
Notation layer: Apostol uses standard 1969-vintage notation ( $a, b$ bold for vectors; $\nabla$ for gradient, $\nabla \cdot$ for divergence, $\nabla \times$ for curl; $det A$ or $∣ A ∣$ ; $i, j, k$ for $R^{3}$ basis). Aligned with modern Codex conventions; no notation/apostol-vol2.md needed if the new units adopt the modern $d f_{p}$ / $J_{f}$ / $det A$ conventions.
Sequencing layer: ~10%. Codex DAG has no linear-algebra spine; the new units must build a fresh DAG segment in 01-foundations and 02-analysis (multivariable-differentiation, ODE).
Intuition layer: ~10%. Apostol's geometric / kinematic intuition for line integrals, gradient flow, eigenvalue interpretation as scaling-along-eigenvector is essentially absent in Codex.
Application layer: ~30%. The vector-calculus applications (work, flux, conservative fields, Maxwell-equation precursors) are partly covered in 08.* stat-mech / lattice-gauge units; the linear-algebra applications (PCA, normal modes via eigenvalues, matrix exponential in dynamics) are absent.

§3 Gap punch-list (P3-lite — units to write or deepen, priority-ordered)

The Codex 01-foundations / 02-analysis chapters need substantial expansion to honestly claim Apostol-Vol-2-equivalence. The bulk of this is [NEW], with [ENRICH] / [DEEPEN] confined to the vector-calculus pieces that already have a manifold-form treatment in 03.04.*. The recommended slot ranges are:

01.01.* for the linear-algebra core (linear maps, determinants, eigenvalues, spectral theorem).
01.01.E* for linear-algebra exercise packs.
02.05.* (new chapter) for multi-variable differential calculus.
02.06.* (new chapter) for ODEs (linear ODEs and systems).
02.10.* (new chapter) for classical vector calculus (line / surface / multiple integrals in classical $\nabla$ -flavoured form).
Existing 03.04.* units for the manifold-form Stokes / Green / divergence enrichments.
Probability + numerical analysis (02.12.*) deferred — see §5.

Priority 1 — Linear-algebra spine and load-bearing multi-variable calculus

These items create the operational vocabulary that the entire rest of Codex silently assumes. Without them, every downstream unit's prerequisites are floating — 03.02.01-smooth-manifold cannot honestly demand "linear map" / "differential" / "Jacobian" because none of those are shipped.

01.01.04 Subspace, basis, dimension. [NEW]. Apostol Ch. 1 §1.10-1.13 anchor; Halmos Finite-Dimensional Vector Spaces (1958) originator-citation. Three-tier; ~1500 words. Master section: span, linear independence, Steinitz exchange lemma, dimension theorem $dim (U + V) + dim (U \cap V) = dim U + dim V$ , dimension as well-defined invariant. Worked examples: $dim R^{n} = n$ , $dim P_{n} [x] = n + 1$ (polynomials of degree $\leq n$ ).
01.01.05 Linear transformation: kernel, image, rank-nullity. [NEW]. Apostol Ch. 2 §2.2-2.7 anchor; Halmos originator. Three-tier; ~1800 words. Master section: linear map $T : V \to W$ between vector spaces; $ker T$ , $im T$ as subspaces; rank-nullity $dim V = dim ker T + dim im T$ ; consequences for injectivity / surjectivity in finite dimensions. Highest priority in the linear-algebra block — every other linear-algebra unit depends on this.
01.01.06 Matrix of a linear map; change of basis; matrix algebra. [NEW]. Apostol Ch. 2 §2.13-2.21 anchor. Three-tier; ~2000 words. Master section: matrix representation in fixed bases; matrix multiplication = composition of linear maps; transpose, inverse, similarity transformation $A \mapsto P^{- 1} A P$ ; change-of- basis formula for endomorphisms. Worked examples: $2 \times 2$ rotation matrix, projection matrix, similarity to diagonal form.
01.01.07 Determinant: axiomatic + expansion + properties. [NEW]. Apostol Ch. 3 anchor; Cayley 1858 A memoir on the theory of matrices originator (for $n \geq 3$ ); Leibniz / Cramer for the classical formula. Three-tier; ~2200 words. Master section: the determinant as the unique multilinear-alternating-normalised function $det : V^{n} \to F$ (where $dim V = n$ ); equivalence with permutation-sum formula $\det A = \sum_\sigma \mathrm{sgn}(\sigma) \prod_i a_{i,\sigma(i)}$ and Laplace expansion; properties ( $det (A B) = det A \cdot det B$ , $det A^{T} = det A$ , behaviour under row operations, Cramer's rule); geometric interpretation as signed volume. Load-bearing — 03.04.02-differential-forms silently uses determinants in the top-form construction.
01.01.08 Eigenvalue, eigenvector, characteristic polynomial. [NEW]. Apostol Ch. 4 §4.1-4.10 anchor; Cauchy 1829 (originator — characteristic equation in mechanics). Three-tier; ~2000 words. Master section: eigenvalue equation $T v = λ v$ ; characteristic polynomial $χ_{T} (λ) = det (λ I - T)$ ; algebraic vs geometric multiplicity; diagonalisability iff sum of geometric multiplicities equals $dim V$ ; Cayley-Hamilton theorem (with proof via adjugate matrix). Worked examples: $2 \times 2$ diagonalisable, $2 \times 2$ non-diagonalisable Jordan block, rotation eigenvalues $e^{\pm i θ}$ .
01.01.09 Inner-product space: orthogonality, Gram-Schmidt, spectral theorem (finite-dim). [NEW]. Apostol Ch. 5 anchor; Hilbert 1904 originator (early form), Schmidt 1907 originator (Gram-Schmidt). Three-tier; ~2200 words. Master section: inner product $⟨ \cdot, \cdot ⟩$ on real / complex vector space; orthonormal basis; Gram-Schmidt orthogonalisation; orthogonal projection; orthogonal / unitary operator preserves $\langle \cdot, \cdot \rangle$; spectral theorem for symmetric / Hermitian operators — every Hermitian operator on a finite-dimensional inner-product space has an orthonormal basis of eigenvectors with real eigenvalues. Cross-link to 02.11.03-unbounded-self-adjoint for the infinite-dimensional generalisation. Load-bearing for 03.05.09-curvature (sectional curvature is an eigenvalue computation), 03.09.* spin-geometry (Dirac operator spectrum), 07.* representation theory (unitary reps).
01.01.10 Bilinear and quadratic forms; Sylvester's law of inertia; principal-axes theorem. [DEEPEN] of 01.01.15-bilinear- quadratic-form. Apostol Ch. 5 §5.13-5.18 anchor; Sylvester 1852 originator. Currently the unit is templated. Replace the templated "Key theorem" section with: Sylvester's law of inertia (signature $(p, q, z)$ is invariant under change of basis); principal-axes theorem (every quadratic form has an orthonormal basis in which it is diagonal); applications to conic and quadric classification. Cross-link to item 6. No new unit ID.
02.05.01 Multi-variable limit and continuity. [NEW]. Apostol Ch. 8 §8.2-8.4 anchor. Three-tier; ~1300 words. Master section: limit of $f : R^{n} \to R^{m}$ at a point; $ϵ$ - $δ$ definition; continuity; sequential continuity; continuity-of-composition. Worked examples: $f(x, y) = xy / (x^2 + y^2)$ has no limit at origin; partial limits don't determine joint limit.
02.05.02 Partial derivative; differentiability; the differential $d f_{p}$ . [NEW]. Apostol Ch. 8 §8.5-8.13 anchor; Cartan 1899-1900 originator (intrinsic differential); Dieudonné 1960 Foundations of Modern Analysis for the modern presentation. Three-tier; ~2000 words. Master section: partial derivative; existence of partials does not imply differentiability; differentiability defined as existence of linear map $d f_{p} : R^{n} \to R^{m}$ with $f (p + h) = f (p) + d f_{p} (h) + o (∣ h ∣)$ ; sufficient condition ( $C^{1}$ implies differentiable); Jacobian matrix as matrix of $d f_{p}$ in standard bases. Load-bearing — 03.02.01-smooth-manifold silently uses this.
02.05.03 Chain rule for multi-variable functions. [NEW]. Apostol Ch. 8 §8.18-8.21 anchor. Three-tier; ~1500 words. Master section: chain rule $d (g \circ f)_{p} = d g_{f (p)} \circ d f_{p}$ ; matrix form $J_{g \circ f} (p) = J_{g} (f (p)) \cdot J_{f} (p)$ ; gradient of composition; directional derivative; consequences. Worked examples: chain rule in polar coordinates, change of variables in PDE.
02.05.04 Implicit and inverse function theorems. [NEW]. Apostol Ch. 9 §9.10-9.16 anchor; Dini 1877 originator (implicit); Goursat 1903 originator (modern statement). Three-tier; ~2400 words. Master section: inverse function theorem — if $f$ is $C^{1}$ near $p$ and $d f_{p}$ invertible, then $f$ is locally invertible with $C^{1}$ inverse; implicit function theorem — derived from inverse function theorem; full proof via Banach fixed-point / contraction mapping. Load-bearing — 03.02.01-smooth-manifold invokes regular-value theorem (= IFT corollary) silently.
02.05.05 Taylor's theorem and extrema in several variables. [NEW]. Apostol Ch. 9 §9.4-9.9 anchor. Three-tier; ~1500 words. Master section: multi-variable Taylor's theorem with remainder; Hessian matrix as second-order coefficient; sufficient conditions for local extremum (positive-definite Hessian = local min); Lagrange multipliers via implicit function theorem (cross-link to item 11). Worked examples: classify critical points of $f (x, y) = x^{2} - y^{2} + x y$ ; constrained optimisation on a sphere.

Priority 2 — ODEs and classical vector calculus

These items open two new chapters in 02-analysis (ODE infrastructure and classical vector calculus) and provide the [ENRICH] hooks into the existing manifold-form Stokes machinery.

02.06.01 First-order linear ODE; existence and uniqueness for linear ODEs. [NEW]. Apostol Ch. 6 §6.4-6.7 anchor. Three-tier; ~1500 words. Master section: $y^{'} + p (x) y = q (x)$ solved via integrating factor; uniqueness of solution given initial condition; statement of Picard-Lindelöf for linear case. Worked examples: radioactive decay, Newton's cooling law.
02.06.02 $n$ -th-order linear ODE with constant coefficients. [NEW]. Apostol Ch. 6 §6.9-6.18 anchor; Euler 1739-1743 (originator — characteristic equation method); d'Alembert 1747 (inhomogeneous case). Three-tier; ~2000 words. Master section: solution space of $L y = 0$ where $L$ is a constant-coefficient differential operator is an $n$ -dimensional vector space; characteristic polynomial of $L$ as the eigenvalue problem of $D = d / d x$ (cross-link to item 5); fundamental solution set; Wronskian as test for linear independence; inhomogeneous case via variation of parameters. Worked examples: harmonic oscillator $\overset{x}{¨} + ω^{2} x = 0$ ; damped oscillator; forced oscillator with resonance.
02.06.03 Systems of linear ODEs and the matrix exponential. [NEW]. Apostol Ch. 7 anchor; Lagrange 1762-1765 (originator, in mechanics); Cauchy 1840s (matrix exponential form). Three-tier; ~2000 words. Master section: system $\overset{x}{˙} = A x$ where $A \in \mathrm{Mat}_n(\mathbb{R}) $; so l u t i o n$ x(t) = e^{tA} x(0)$; matrix exponential defined via power series; computation via diagonalisation when $A$ diagonalisable; via Jordan form otherwise; phase-portrait classification (sink / source / saddle / centre / spiral) by eigenvalue location. Load-bearing — 03.03.01-lie- group invokes the matrix exponential silently.
02.10.01 Line integral; conservative fields; path independence. [NEW]. Apostol Ch. 10 §10.1-10.18 anchor. Three-tier; ~1800 words. Master section: line integral $\int_\gamma P,dx + Q,dy + R,dz $a l o n g a p i ece w i se -$ C^1 $c u r v e$ \gamma$; line integral of vector field $F \cdot d r$ ; conservative field $F = \nabla ϕ$ ; path independence iff conservative iff $\oint = 0$ on every closed loop; relation to closed-vs-exact 1-forms (cross-link to 03.04.06-de-rham- cohomology). Worked examples: work done by gravity (conservative); work in magnetic field (not conservative).
02.10.02 Multiple integral; Riemann integrability over a rectangle and over a Jordan-measurable set. [NEW]. Apostol Ch. 11 §11.2-11.10 anchor; Riemann 1854 Habilitationsschrift originator-citation. Three-tier; ~1600 words. Master section: Riemann integral over a rectangle in $R^{n}$ ; Jordan content / Jordan-measurable sets; integrability of continuous functions; properties (linearity, monotonicity, additivity over disjoint decomposition). Bridge to 02.11.* measure theory (deferred).
02.10.03 Fubini's theorem (Riemann form) and iterated integrals. [NEW]. Apostol Ch. 11 §11.7 anchor. Three-tier; ~1200 words. Master section: continuous-integrand Fubini — $\int_{R_{1} \times R_{2}} f = \int_{R_{1}} (\int_{R_{2}} f)$ when $f$ continuous; equivalent to Riemann definition. Cross-link to measure-theoretic Fubini (deferred).
02.10.04 Change of variables in multiple integrals; the Jacobian formula. [NEW]. Apostol Ch. 11 §11.26-11.34 anchor; Jacobi 1841 De determinantibus functionalibus originator. Three-tier; ~1700 words. Master section: change-of-variables formula $\int_{Ω} f = \int_{Ω^{'}} (f \circ g) \cdot ∣ det D g ∣$ for $C^{1}$ diffeomorphism $g$ ; proof sketch via partition / linear approximation; standard worked examples — polar, cylindrical, spherical coordinates; relation to differential-forms version (cross-link to 03.04.03-integration-on-manifolds). Load-bearing operationally — used silently in physics applications throughout Codex.
02.10.05 Surface integral and parametric surfaces. [NEW]. Apostol Ch. 12 §12.1-12.10 anchor. Three-tier; ~1500 words. Master section: parametric surface $r : D \to R^{3}$ ; surface area $\iint_S dS = \iint_D |\mathbf{r}_u \times \mathbf{r}_v| , du , dv $; f l ux in t e g r a l$ \iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F} \cdot (\mathbf{r}_u \times \mathbf{r}_v) , du , dv$; relation to integration of 2-forms (cross-link to 03.04.03).
02.10.06 Vector calculus identities; gradient / divergence / curl in $R^{3}$ . [NEW]. Apostol Ch. 12 §12.16-12.18 anchor. Three-tier; ~1300 words. Master section: classical $\nabla$ operator; $\nabla \times \nabla f = 0$ , $\nabla \cdot \nabla \times \mathbf{F} = 0$; vector triple identities; classical-vs-modern bridge translating $(\nabla, \nabla \cdot, \nabla \times)$ to $d$ on differential forms via the musical isomorphism in $R^{3}$ (cross-link to 03.04.04-exterior-derivative). Pedagogically important — closes the gap between classical-physics-vector-calculus notation and the modern differential-forms language.
[ENRICH] 03.04.05-stokes-theorem. Add Master-section treatment of the three classical specialisations: Green's theorem in the plane ( $n = 2$ ), classical Stokes (boundary of a 2-surface in $R^{3}$ ), divergence theorem ( $n = 3$ , $\nabla \cdot \mathbf{F}$ form). With Apostol Ch. 12 anchor. Each as a one-page Master subsection deriving the classical statement from the abstract manifold-form Stokes. No new unit ID.
[ENRICH] 03.04.03-integration-on-manifolds. Add cross-link paragraph to 02.10.04 and 02.10.05 for the classical-form treatments. Note that the change-of-variables formula in classical notation is the differential-forms pull-back formula in disguise. No new unit ID.
[ENRICH] 03.04.06-de-rham-cohomology. Add cross-link to 02.10.01 (conservative fields) — closed-vs-exact 1-forms in $R^{n}$ correspond exactly to conservative-vs-non-conservative vector fields; on a star-shaped domain, every closed 1-form is exact, recovering Poincaré's lemma in the classical-vector-calculus setting.

Priority 3 — Linear-algebra depth completions and exercise packs

These items close the remaining linear-algebra depth gaps and produce the (large) exercise pack required for full equivalence.

01.01.11 Jordan canonical form and minimal polynomial. [NEW]. Apostol Ch. 4 §4.20 + Hoffman-Kunze for full proof. Three-tier; ~1800 words. Master section: minimal polynomial; primary decomposition theorem; Jordan blocks; Jordan canonical form theorem (every endomorphism over algebraically closed field has Jordan form, unique up to ordering); applications to matrix exponential (cross-link to 02.06.03). Optional but valuable.
01.01.12 Singular value decomposition (finite-dim). [NEW]. Beltrami 1873 / Jordan 1874 / Sylvester 1889 originator. Three-tier; ~1500 words. Apostol does not develop SVD per se but the principal- axes theorem (item 7) is the symmetric case of the same idea; SVD is a natural priority-3 add-on for completeness. Worked example: PCA / data-analysis interpretation.
01.01.E1 Linear-algebra exercise pack: matrix computations, rank-nullity, eigenvalues, diagonalisation. [NEW]. Apostol's Ch. 1-5 exercise selection. ~50 problems with full or hint-form solutions. Operational competence test.
02.05.E1 Multi-variable calculus exercise pack. [NEW]. Apostol's Ch. 8-9 exercise selection. ~30 problems on partial derivatives, chain rule, IFT applications, Lagrange multipliers, Taylor expansions.
02.06.E1 ODE exercise pack. [NEW]. Apostol's Ch. 6-7 exercise selection. ~20 problems on linear ODEs, characteristic polynomials, matrix exponential.
02.10.E1 Vector calculus exercise pack. [NEW]. Apostol's Ch. 10-12 exercise selection. ~30 problems on line integrals, multiple integrals (incl. polar/cylindrical/spherical), surface integrals, applications of Green / Stokes / divergence theorems.
[DEEPEN] 01.01.03-vector-space. Replace templated v0.5 prose with anchored treatment: explicitly cite Halmos and Apostol; add Master-section discussion of finite-dimensional vs infinite-dimensional distinction; cross-link to item 1. No new unit ID.

Priority 4 — Probability and numerical analysis (deferred / out of scope)

These items are deliberately deferred — probability theory belongs in a measure-theory chapter that does not yet exist; numerical analysis is out of strict FT scope. Listed for completeness:

02.12.* Elementary probability (Apostol Ch. 13-15). [DEFERRED]. Defer until measure-theory chapter exists in 02.*. A future plan can cover Apostol's $σ$ -algebra-via-set-functions treatment as the bridge to formal measure theory. Out of scope for the FT spine; revisit when probability theory is added per the strand expansion plan.
Numerical analysis (Apostol Ch. 16). [OUT OF SCOPE]. Codex deliberately does not cover numerical analysis. A future Numerical Analysis Strand could address Lagrange interpolation, Newton-Cotes, Newton's method; not required for FT equivalence.

§4 Implementation sketch (P3 → P4)

Minimum Apostol-Vol-2-equivalence batch = priority 1 only (items 1-12): 11 new units + 1 deepening across 01.01.04-12 (linear-algebra spine) and 02.05.01-05 (multi-variable differentiation). Realistic production estimate (mirroring earlier Donaldson / Cannas / Hatcher batches):

~3 hours per typical new unit (research + draft + validate at 27/27 + Lean stub + Bridge / Synthesis prose).
~4 hours for the larger units (determinant axioms, IFT, spectral theorem, change-of-variables — each load-bearing for downstream strands).
~1.5 hours per deepening.
Priority-1 totals: 5 large × 4 h + 6 typical × 3 h + 1 deepening × 1.5 h = ~39.5 hours.
Priority-1+2 totals: 5 large × 4 h + 18 typical × 3 h + 4 enrichments × 1 h = ~78 hours.
Priority-1+2+3 (excluding exercise packs): ~95 hours.
Exercise packs (items 27-30): ~3 hours each × 4 = ~12 hours.
Grand total for full Apostol-Vol-2-equivalence (excluding deferred probability and out-of-scope numerical analysis): ~107 hours.

At 6-8 production agents in parallel, priority-1 fits in a 4-5 day window with 1-2 integration agents stitching outputs. This is the largest single Tier-α book production batch in the campaign.

Batch structure.

Batch A (linear-algebra spine, items 1-7, ~28 h): opens out 01.01-linear-algebra/ to ~10 units. Pure prerequisite; produce first; load-bearing for everything downstream including most Tier-β books.
Batch B (multi-variable differential calculus, items 8-12, ~16 h): opens new chapter 02.05-multivariable-differentiation/. Depends on Batch A only for the linear-algebra prerequisites (linear map, matrix, determinant).
Batch C (ODEs, items 13-15, ~12 h): opens new chapter 02.06-ordinary-differential-equations/. Depends on Batch A (eigenvalues for matrix exponential, Jordan form for non- diagonalisable case). Bridges to Arnold ODEs (deferred).
Batch D (classical vector calculus, items 16-21, ~14 h): opens new chapter 02.10-vector-calculus/. Depends on Batch B (Jacobian / chain rule for change-of-variables formula). Mostly independent of Batch C.
Batch E (manifold-form enrichments, items 22-24, ~3 h): enriches existing 03.04.* units. Single-threaded; do at end of Batches B-D.
Batch F (exercise packs, items 27-30, ~12 h): depends on A-D; produced in parallel after the theorem-layer batch lands.

Originator-prose targets (each priority-1 unit's Master section cites originator + Apostol):

Linear spaces / dimension (1): Halmos 1958 Finite-Dimensional Vector Spaces; Grassmann 1844 Ausdehnungslehre for the originator- text on linear independence and dimension as an algebraic-geometric concept.
Linear maps / rank-nullity (2): Halmos 1958; Sylvester 1851 for "rank" vocabulary.
Matrix algebra (3): Cayley 1858 A memoir on the theory of matrices; Frobenius 1877 (similarity / canonical forms).
Determinants (4): Leibniz 1693 (originator concept); Cauchy 1812 (modern formulation); Cayley 1841 (notation $det$ ).
Eigenvalues (5): Cauchy 1829 (characteristic equation in mechanics); Cayley-Hamilton 1853-1858.
Inner-product / spectral theorem (6): Hilbert 1904 (early form); Schmidt 1907 (Gram-Schmidt); von Neumann 1929-32 Mathematische Grundlagen der Quantenmechanik (modern axiomatic).
Multi-variable differential (9): Cartan 1899-1900 (intrinsic differential); Dieudonné 1960 Foundations of Modern Analysis.
IFT / inverse function theorem (11): Dini 1877; Goursat 1903.
Linear ODE (14): Euler 1739-1743; d'Alembert 1747.
Matrix exponential / systems (15): Lagrange 1762-1765; Peano 1888 (rigorous matrix exponential).
Multiple integral / Fubini (17-18): Riemann 1854; Fubini 1907; Tonelli 1909.
Change of variables (19): Jacobi 1841 De determinantibus functionalibus.

Notation crosswalk. Apostol's notation is mostly aligned with modern Codex conventions ( $R^{n}$ , $\nabla f$ , $J_{f}$ , $det A$ , $⟨ \cdot, \cdot ⟩$ ). Distinctive: bold $a$ for vectors (Codex prefers italic $a$ except in physics contexts); $∣ A ∣$ sometimes interchangeable with $det A$ . No notation/apostol-vol2.md strictly needed; alignment can be handled in each new unit's notation discussion.

DAG edges to add. New prerequisites for the priority-1+2 batch:

01.01.04 ← 01.01.03-vector-space
01.01.05 ← 01.01.04; → 03.02.01-smooth-manifold (tangent map); → 02.05.02 (differential as linear map)
01.01.06 ← 01.01.05
01.01.07 (determinants) ← 01.01.06; → 03.04.02-differential-forms (top form construction); → 01.01.08
01.01.08 (eigenvalues) ← {01.01.07, 01.01.06}; → 02.06.02-linear-ode-constant-coefficients; → 02.06.03-systems-matrix-exponential; → 02.11.03-unbounded-self-adjoint; → 07.04.01-cartan-weyl-classification
01.01.09 (inner-product / spectral) ← {01.01.08, 02.11.07-inner-product-space}; → 01.01.15-bilinear-quadratic-form; → 03.05.09-curvature (sectional curvature); → 03.09.08-dirac-operator
02.05.01 (limits) ← 01.01.05 (linear map context)
02.05.02 (differential) ← {02.05.01, 01.01.05, 01.01.06}; → 03.02.01-smooth-manifold
02.05.03 (chain rule) ← 02.05.02
02.05.04 (IFT / inverse function theorem) ← {02.05.03, 02.11.04-banach-spaces for Banach fixed-point}; → 03.02.01-smooth-manifold (regular-value theorem)
02.05.05 (Taylor / extrema) ← {02.05.03, 01.01.09 for Hessian spectral analysis}
02.06.01-03 (ODEs) ← 01.01.08
02.10.01 (line integral) ← 02.05.03
02.10.02-04 (multiple integral / Fubini / change of variables) ← {02.05.03, 01.01.07-determinants}
02.10.04 → 03.04.03-integration-on-manifolds (the modern reformulation)
02.10.05-06 (surface integral / vector calculus identities) ← {02.10.04, 03.04.04-exterior-derivative for the bridge}

Composite Apostol-Vol-2 + Shilov batch recommendation. Shilov Linear Algebra (FT 1.01) is Tier-α and shares the linear-algebra spine with Apostol Vol 2 Chapters 1-5 entirely. Items 1-7 of this plan also constitute the bulk of the Shilov priority-1 batch. Defer the Shilov audit to the same campaign window so that these heavy units are produced once and serve both equivalences. Hoffman-Kunze and Lang Algebra (Tier-α 3.01) overlap with item 25 (Jordan form) and items 1-6 — produce in synchronicity.

Composite with Apostol Vol 1 (FT 0.2): Apostol Vol 1 covers single-variable calculus and would supply prerequisites in 00-precalc / 02.0X-single-variable-analysis. Vol 1 and Vol 2 should be planned adjacent: the §0 prerequisites slot is a single coherent foundational batch.

§5 What this plan does NOT cover

Probability theory (Apostol Ch. 13-15). Deferred until a measure-theory chapter exists in 02.*. Apostol's elementary-form treatment (set-function-based, Kolmogorov axioms, discrete + continuous random variables, expectation, Chebyshev, weak law of large numbers) is a self-contained pre-measure-theoretic introduction; it is left P4 / out of scope here. Revisit when probability theory enters Codex via the Diaconis (Tier-γ 3.16) or Tao Measure Theory expansion.
Numerical analysis (Apostol Ch. 16). Deliberately out of scope for the FT spine. Codex covers no numerical analysis; Lagrange interpolation, Newton-Cotes, Newton's method, fixed-point iteration are off-curriculum. Defer indefinitely.
Apostol Vol 1 material (single-variable calculus, $ϵ$ - $δ$ rigorous limits, Riemann integral on $R$ , series, Taylor's theorem in one variable). Vol 1 is FT 0.2 and gets its own audit; some Vol 2 units (02.05.01 limits, 02.05.05 Taylor) reference Vol 1 prerequisites.
Line-number-level inventory of every theorem / exercise across Apostol Vol 2's 800 pages. Defer unless priority-1+2 expands. The ~2400 exercises require dedicated exercise-pack production — items 27-30 are placeholder packs of ~30-50 problems each, not full exercise reproduction.
General Picard-Lindelöf existence-uniqueness for nonlinear ODEs (Apostol's Ch. 6 only treats the linear case rigorously; the general case is in Arnold ODEs and Coddington-Levinson). Defer to Arnold ODEs batch.
Full Jordan-form proof (Apostol gives partial; Hoffman-Kunze / Lang fill it). Item 25 is priority-3 / optional.
Functional analysis (Banach / Hilbert space theory in infinite dimensions). Codex's 02.11.* already covers this, at greater depth than Apostol Vol 2 reaches. No deepening required on that side.
Differential forms / manifold-form Stokes (Apostol Ch. 12 stays classical; the modern manifold treatment is 03.04.* and is already shipped). Items 22-24 are [ENRICH]-only cross-link additions, not new manifold-side content.
Tensor analysis / curvilinear coordinates (Apostol Ch. 12 appendix touches this; full treatment is in Sternberg / Frankel / Lovelock-Rund — not Apostol's central scope).
Modern multi-variable analysis beyond Apostol's 1969 vintage (Lebesgue integral, distributions, Sobolev spaces). Out of scope; in 02.11.* / future measure-theory chapter.

§6 Acceptance criteria for FT equivalence (Apostol Vol 2)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 and §9, Apostol Vol 2 is at equivalence-coverage when:

≥95% of Apostol Vol 2's named theorems (in Chapters 1-12) map to Codex units at Apostol-equivalent proof depth (currently ~15%; after priority-1 this rises to ~55%; after priority-1+2 to ~80%; after priority-3 deepenings to ~92% — modulo deliberate-out-of-scope Chapters 13-16).
≥80% of Apostol Vol 2's exercises (Chapters 1-12) have a Codex equivalent. This is the largest single exercise-layer gap on the Fast Track. Currently ~1%; closing this requires items 27-30 (priority-3 exercise packs) plus ongoing per-unit exercise densification. Realistic path-to-≥80% needs an explicit follow-up Apostol-Vol-2-exercise-pack pass beyond items 27-30.
≥90% of Apostol Vol 2's worked examples (Chapters 1-12) are reproduced in some Codex unit. Currently ~5%; the priority-1+2 batch's mandatory worked-example inclusions (matrix computations, Jacobian computations, ODE solutions, change-of-variables in polar / cylindrical / spherical, Lagrange-multiplier examples) bring this to ~70%; densification to ~90% is part of the per-unit production spec, not a separate task.
Notation alignment recorded inline in the new units (no separate notation/apostol-vol2.md strictly required; alignment with modern conventions is the default).
For every chapter dependency in Apostol Vol 2 (Ch. 1 → Ch. 2 → Ch. 3 → Ch. 4 → Ch. 5; Ch. 5 → Ch. 6 → Ch. 7; Ch. 1-5 → Ch. 8-9; Ch. 8-9 → Ch. 10-12), there is a corresponding prerequisites arrow chain in Codex's DAG. The linear-algebra → multi-variable- differentiation → vector-calculus chain in particular must be unbroken after the priority-1+2 batch.
Pass-W weaving connects the new units (01.01.04-12, 02.05.01-05, 02.06.01-03, 02.10.01-06) to the existing 03.02.01-smooth-manifold, 03.04.* differential-forms units, and the 02.11.* functional-analysis chapter via lateral connections. The classical-vs-modern bridge (item 21 + items 22-24) is the single most pedagogically important weaving lateral.

The 12 priority-1 items (items 1-12) close the load-bearing-linear- algebra-and-multi-variable-differentiation gap that every other Codex strand silently assumes. Priority-2 items (13-24) open the ODE and classical-vector-calculus chapters and provide the [ENRICH] hooks into the existing manifold-form Stokes machinery. Priority-3 items (25-31) are linear-algebra depth completions plus the (large) exercise pack. Priority-4 items (32-33) are deferred / out-of-scope.

Honest scope. The largest single Tier-α equivalence gap. Codex 01-foundations / 02-analysis is structurally undersized relative to the foundational role these chapters play in every downstream strand; Apostol Vol 2 production is prerequisite work for the whole Fast Track campaign. Once shipped, downstream books (Hatcher, Bott-Tu, Sternberg, Lawson-Michelsohn, Cannas, Donaldson, Arnold, Lang, Shilov) all gain explicit linear-algebra and multi-variable-calculus prerequisites instead of silent assumptions.

Largest single Apostol-Vol-2-distinctive value-add: the linear-algebra spine (items 1-7). Codex has no rank-nullity, no matrix algebra, no determinants, no eigenvalues, no spectral theorem as named units. Producing them is a one-time foundational investment with the highest leverage of any priority-1 batch in the campaign.

Composite recommendation. Plan and produce **Apostol Vol 1 + Vol 2

Shilov + Lang Algebra** as a synchronised §0 / §1 prerequisite- foundation batch. The four books overlap heavily on linear-algebra and single/multi-variable analysis; producing the foundational units once and reusing across all four equivalence sign-offs is the correct campaign sequencing. After that joint foundation lands, all subsequent Tier-α / β audits can drop their "linear algebra silently assumed" caveats and demand named-unit prerequisites.

Apostol — *Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability* (Fast Track 0.3) — Audit + Gap Plan