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Curriculum & Standards · Calculus II / Multivariable

Calculus II / Multivariable — the bridge from freshman calculus to STEM major, calibrated to MIT 18.02 + 18.03 + Berkeley Math 53/54 + Stanford + Cambridge Tripos + Tokyo + UNAM.

From 3D vector geometry with cross product and scalar triple product, to vector-valued functions with Frenet trihedron and curvature, to multivariable limits with the path test, to partial derivatives with Clairaut's theorem and the multivariable chain rule, to gradient + directional derivative + the second-derivative test in 2D + Lagrange multipliers (one and two constraints), to double and triple integrals in rectangular / polar / cylindrical / spherical coordinates with Jacobians and applications, to vector fields with divergence and curl, line integrals with the Fundamental Theorem and Green's theorem, surface integrals with Stokes' theorem and the divergence theorem, and finally to ODEs II — linear first-order with integrating factor + exact + Bernoulli + first-order modeling + second-order linear constant-coefficient with the three classical regimes (real distinct, repeated, complex) + undetermined coefficients with resonance modification + variation of parameters with Wronskian + reduction of order + mass-spring (unforced, damped, forced + resonance) + RLC circuit. This is the FIRST Rodybee program where standard HS exam systems (AP, IB, Cambridge A-level core) do NOT cover the material — the benchmark is fully university.

98

Skills

14

Modules (hex)

7

Official standards benchmarked

18–20

Age coverage

Why Calculus II / Multivariable is essential for any STEM major

Multivariable calculus + vector calculus + ODEs II is the universal second-year math at every research university: MIT, Stanford, Berkeley, Cambridge, Oxford, Tokyo, UNAM. Without partial derivatives, gradients, Stokes/divergence theorems and second-order ODEs you cannot do classical mechanics, electromagnetism, fluid dynamics, thermodynamics, signal processing, optimization, mathematical biology, mathematical economics, or modern data science. AP/IB/Cambridge A-level do NOT cover it; this is where Rodybee's curriculum joins the university freshman/sophomore math sequence at the world's top institutions.

#CountryQS Math rank
1🇺🇸MIT99
2🇬🇧Cambridge96
3🇬🇧Oxford95
4🇺🇸Stanford94
5🇺🇸UC Berkeley93

Source: QS World University Rankings — Mathematics 2024 ↗

Find your child's level

Pick an age and see exactly what we teach at that point — and what the most demanding standards in the world expect at the same age. There's no marketing fluff: each topic links to the official curriculum document.

Your child's age

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What we teach at this age

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Highest international bar at this age · Univ Freshman Sem 1

🇺🇸Estados Unidos

3D coordinate geometry deep (full dot product, cross product, scalar triple product, vector projection, lines and planes in 3D, distance from point to line / plane, skew lines), six canonical quadric surfaces (ellipsoid, elliptic + hyperbolic paraboloid, hyperboloid one + two sheets, cone, cylinder), vector-valued functions r(t) with derivative and integral, unit tangent T, principal normal N, binormal B (Frenet trihedron), arc length parametrization, curvature κ, torsion τ (intuitive), motion decomposition into tangential + normal acceleration, multivariable functions with domain/level curves/level surfaces, ε-δ multivariable limits, path test for non-existence, iterated limits distinction, multivariable continuity, partial derivatives + geometric interpretation + higher-order + Clairaut's theorem, differentiability multivariable, tangent plane equation, linear approximation + total differential, multivariable chain rule (one + two independent + tree diagrams), implicit differentiation 2D/3D, directional derivative, gradient ∇f and its perpendicular relationship to level curves/surfaces.

Grade-by-grade benchmark

Full table of what each major standard expects per grade. Every cell is sourced from the official ministry, board or framework document.

Univ Freshman Sem 1

First third — 3D geometry + vector functions + multivariable diff calculus (~18–19 años)

Highest bar · 🇺🇸 Estados Unidos

3D coordinate geometry deep (full dot product, cross product, scalar triple product, vector projection, lines and planes in 3D, distance from point to line / plane, skew lines), six canonical quadric surfaces (ellipsoid, elliptic + hyperbolic paraboloid, hyperboloid one + two sheets, cone, cylinder), vector-valued functions r(t) with derivative and integral, unit tangent T, principal normal N, binormal B (Frenet trihedron), arc length parametrization, curvature κ, torsion τ (intuitive), motion decomposition into tangential + normal acceleration, multivariable functions with domain/level curves/level surfaces, ε-δ multivariable limits, path test for non-existence, iterated limits distinction, multivariable continuity, partial derivatives + geometric interpretation + higher-order + Clairaut's theorem, differentiability multivariable, tangent plane equation, linear approximation + total differential, multivariable chain rule (one + two independent + tree diagrams), implicit differentiation 2D/3D, directional derivative, gradient ∇f and its perpendicular relationship to level curves/surfaces.

CountryExpected topicsOfficial source
🇺🇸Estados UnidosMIT 18.02 §1 (vectors, lines, planes, quadrics), §13 (vector functions, Frenet, curvature), §14.1–14.6 (multivariable functions, partials, gradient, chain rule).MIT OCW 18.02 Multivariable Calculus ↗
🇺🇸Estados UnidosUC Berkeley Math 53: 3D geometry, partial derivatives, gradient, chain rule, directional derivative — primer semestre.UC Berkeley Math 53 (Multivariable) + Math 54 (LA + ODEs) ↗
🇺🇸Estados UnidosStanford MATH 51 (Multivariable Calculus): vectores 3D, parciales, gradiente, derivadas direccionales.Stanford MATH 51 / 52 / 53 (Multivariable + Vector Calc + ODEs) ↗
🇲🇽MéxicoUNAM Cálculo III: geometría analítica 3D, funciones de varias variables, parciales, regla de la cadena, gradiente.UNAM Cálculo Diferencial e Integral III + IV (Facultad de Ciencias) ↗
🇬🇧Reino UnidoCambridge Tripos IA Vector Calculus §1 (vectores 3D, geometría), §2 (curvas y campos escalares).Cambridge Mathematical Tripos Part IA — Vector Calculus + Differential Equations ↗
🇯🇵Japón東京大学 解析学 II §1 (3D geometría) + §2 (curvas vectoriales con T/N/B y curvatura).東京大学 解析学 II (Tokyo University — Multivariable Analysis II) ↗

Rodybee skills at this grade

3D vectors (full review) · Dot product 3D (full) · Cross product u × v · Scalar triple product · Vector projection in 3D · Lines in 3D (parametric + symmetric) · Planes in 3D (normal + scalar form) · Distance — point/line/plane in 3D · Quadric — ellipsoid · Quadric — paraboloid (elliptic + hyperbolic) · Quadric — hyperboloid (1 + 2 sheets) · Quadric — cone + cylinder · Identify a quadric from equation · Vector-valued function r(t) · Derivative and integral of r(t) · Unit tangent T and principal normal N · Binormal B and Frenet trihedron · Arc-length parametrization · Curvature κ (full formulas) · Torsion τ (intuitive) · Motion: tangential + normal acceleration · Multivariable function domain · Level curves and level surfaces · Multivariable ε-δ limit · Multivariable limit — path test · Iterated vs simultaneous limits · Multivariable continuity · Partial derivative — definition · Partial derivative — geometric meaning · Higher-order partial derivatives · Clairaut's theorem (mixed partials) · Differentiability (multivariable) · Tangent plane equation · Linear approx + total differential · Chain rule (one independent variable) · Chain rule (two independent variables) · Chain rule via tree diagrams · Implicit differentiation 2D and 3D · Directional derivative D_u f · Gradient vector ∇f · ∇f perpendicular to level curves/surfaces

Univ Freshman Sem 2 / Sophomore Sem 1

Second third — extremos, integrales múltiples + cálculo vectorial completo (~18–20 años)

Highest bar · 🇺🇸 Estados Unidos

Multivariable extrema (critical points, second derivative test 2D with discriminant D, saddle points, absolute extrema on closed bounded regions), Lagrange multipliers (one + two constraints), Taylor 2D quadratic with Hessian. Double integrals (rectangles, Type I, Type II, change of order, polar coordinates, applications: volume / mass / centroid / average value). Triple integrals (rectangular, cylindrical x = r cos θ + y = r sin θ + z = z, spherical x = ρ sin φ cos θ + …), Jacobian for general change of variables, volume / mass / center of mass / moments of inertia. Vector fields with divergence ∇·F (sources/sinks) + curl ∇×F (rotation), conservative fields with curl test, finding potential functions, vector field visualization (flow lines). Line integrals scalar (∫_C f ds for wire mass) + vector (∫_C F·dr for work), path independence in conservative fields, Fundamental Theorem for Line Integrals. Green's theorem (circulation form ∮P dx + Q dy and flux form ∮F·n ds), parametric surfaces with normal vector r_u × r_v + orientation, surface integrals (scalar ∫∫ f dS + flux ∫∫ F·dS), Stokes' theorem (3D generalization of Green), divergence theorem (Gauss) for closed surfaces, theorem strategy choice.

CountryExpected topicsOfficial source
🇺🇸Estados UnidosMIT 18.02 §14.7–14.8 (extrema, Lagrange), §15 (integrales múltiples), §16 (cálculo vectorial completo: Green, Stokes, divergencia).MIT OCW 18.02 Multivariable Calculus ↗
🇺🇸Estados UnidosUC Berkeley Math 53: extremos, Lagrange, integrales dobles + triples + vectoriales, Stokes, Gauss.UC Berkeley Math 53 (Multivariable) + Math 54 (LA + ODEs) ↗
🇺🇸Estados UnidosStanford MATH 52: vector calculus completo (Green, Stokes, divergencia) — segundo curso.Stanford MATH 51 / 52 / 53 (Multivariable + Vector Calc + ODEs) ↗
🇲🇽MéxicoUNAM Cálculo III/IV: extremos multivariable, integrales múltiples, cálculo vectorial completo.UNAM Cálculo Diferencial e Integral III + IV (Facultad de Ciencias) ↗
🇬🇧Reino UnidoCambridge Tripos IA Vector Calculus §3 (integrales múltiples), §4 (Green/Stokes/divergencia).Cambridge Mathematical Tripos Part IA — Vector Calculus + Differential Equations ↗

Rodybee skills at this grade

Critical points (multivariable) · Second derivative test 2D (D = f_xx f_yy − f_xy²) · Saddle point classification · Absolute extrema on closed bounded regions · Lagrange multipliers (one constraint) · Lagrange multipliers (two constraints) · Taylor 2D order 2 (Hessian) · Double integral over rectangle (Fubini) · Double integral — Type I region · Double integral — Type II region · Change order of integration · Double integral in polar (r dr dθ) · Double integral — volume application · Double integral — mass + centroid · Triple integral (rectangular) · Triple integral over general 3D region · Triple integral (cylindrical, dV = r dz dr dθ) · Triple integral (spherical, dV = ρ² sin φ dρ dφ dθ) · Jacobian for change of variables · Triple integral — volume + mass · Moments of inertia in 3D · Vector field F(x,y,z) · Divergence ∇ · F · Curl ∇ × F · Conservative field test (curl = 0) · Find potential f with ∇f = F · Visualize vector fields (flow lines) · Scalar line integral ∫_C f ds · Vector line integral ∫_C F · dr (work) · Path independence · FTC for line integrals (gradient theorem) · Green's theorem — circulation form · Green's theorem — flux form · Parametric surface r(u,v) · Surface normal r_u × r_v + orientation · Scalar surface integral ∫∫_S f dS · Flux integral ∫∫_S F · dS · Stokes' theorem · Divergence theorem (Gauss) · Choosing Green vs Stokes vs divergence

Univ Sophomore Sem 1 / Sem 2

Final third — ODEs II completo (~19–20 años)

Highest bar · 🇺🇸 Estados Unidos

Linear first-order ODEs with integrating factor μ(x) = e^∫P dx, exact ODEs M dx + N dy = 0 with test M_y = N_x and potential function Ψ, integrating factors for non-exact, Bernoulli equation y' + Py = Q y^n via substitution u = y^(1−n), first-order modeling (mixing problems, RC circuits, decay). Second-order linear with constant coefficients ay'' + by' + cy = 0: characteristic equation with three regimes (real distinct roots → C₁ e^(r₁x) + C₂ e^(r₂x); repeated root → adds factor x; complex roots α ± βi → e^(αx)(C₁ cos βx + C₂ sin βx)). Non-homogeneous via undetermined coefficients (polynomial guess, exp/trig guess) with resonance modification (multiply by x or x²), variation of parameters using the Wronskian W(y₁, y₂), reduction of order from a known solution. Modeling: mass-spring system unforced (simple harmonic motion ω = √(k/m)), damped (overdamped + critically damped + underdamped regimes), forced with resonance, RLC circuit as electrical analog (m ↔ L, c ↔ R, k ↔ 1/C).

CountryExpected topicsOfficial source
🇺🇸Estados UnidosMIT 18.03 §1 (lineales 1er orden, exactas, Bernoulli), §2–4 (segundo orden constante, undetermined, variación de parámetros, modeling spring/RLC).MIT OCW 18.03 Differential Equations ↗
🇺🇸Estados UnidosUC Berkeley Math 54: ODE cap. 1 (lineales), cap. 2–4 (segundo orden + Wronskiano + modeling).UC Berkeley Math 53 (Multivariable) + Math 54 (LA + ODEs) ↗
🇺🇸Estados UnidosStanford MATH 53 (ODEs): lineales 1er orden + exactas + segundo orden con coeficientes constantes + modeling.Stanford MATH 51 / 52 / 53 (Multivariable + Vector Calc + ODEs) ↗
🇲🇽MéxicoUNAM Cálculo IV cap. 1 (ODEs 1er orden) + cap. 2 (ODEs 2do orden + modeling).UNAM Cálculo Diferencial e Integral III + IV (Facultad de Ciencias) ↗
🇬🇧Reino UnidoCambridge Tripos IA Differential Equations: lineales, exactas, Bernoulli, segundo orden constante, Wronskiano, modeling.Cambridge Mathematical Tripos Part IA — Vector Calculus + Differential Equations ↗

Rodybee skills at this grade

Linear 1st order ODE (integrating factor) · Exact ODE — test and solve · Integrating factor for non-exact ODE · Bernoulli equation · 1st order modeling (mixing, RC) · 2nd order homogeneous — real distinct roots · 2nd order homogeneous — repeated root · 2nd order homogeneous — complex roots · Undetermined coefficients — polynomial g · Undetermined coefficients — exp/trig g · Resonance modification (multiply by x) · Variation of parameters (Wronskian) · Reduction of order · Mass-spring unforced (SHM) · Mass-spring damped (3 regimes) · Mass-spring forced + resonance · RLC circuit (electrical analog)

Full research document

Methodology, gap analysis, recommendations and the complete list of sources are in the underlying research markdown.

Read the full benchmark document →
Calculus II / Multivariable Curriculum & Standards — Rodybee