MAST20029 Engineering Mathematics
The Complete Exam Bible · S2 2026MAST20029 Overview
- University of Melbourne
- S2 2026
- 12 chapters
- Science
Engineering Mathematics (MAST20029) is the University of Melbourne's core second-year mathematics subject for engineering — the gateway that every Engineering Systems major and Master of Engineering student passes through.
- Assessed by Final exam 65% · Mid-semester test 15% · Assignments (3 × 5%) 15% · Quizzes (best 5 of 6) 5% — Mid-semester test, Assignments (3 × 5%), Quizzes (best 5 of 6) are a hurdle
- Key terms Divergence and curl, Jacobian, Conservative field, Critical point classification
- How to prepare Treat MAST20029 as six separate skills that share one exam habit, not one giant subject.
- Most asked Is there a hurdle in MAST20029?
What MAST20029 covers
The whole subject → one exam-ready map. Each chapter links to its free guide; together they mirror the 6 official sections and the 11-question, 120-mark final.
Vector Fields, Divergence and Curl
Section 1 · L1. Vector fields F = F₁ i + F₂ j + F₃ k, the del operator ∇, divergence (scalar), curl (vector via the determinant), gradient, and the identities curl(grad φ)=0 and div(curl F)=0.02Double Integrals and Change of Order
Section 1 · L2. Double integrals over general type-I/II regions, re-describing a region, and changing the order of integration to make an integrand doable — the MST signature skill.03Polar, Cylindrical and Spherical Coordinates
Section 1 · L3–L5. 2D/3D change of variables and the Jacobian: polar (J=r), cylindrical (J=r), spherical (J=r²sin φ) in the course's x=r cos θ sin φ convention; triple integrals.04Line, Work and Surface Integrals; Conservative Fields
Section 1 · L6–L9. Parametrising paths, scalar line and work integrals, conservative fields (curl F=0 ⇒ F=grad φ, path independence, zero loop work), and scalar surface and flux integrals.05Gauss' Divergence Theorem
Section 1 · L10. The divergence theorem ∭ ∇·F dV = ∬ F·n̂ dS over a closed surface with outward normal — converting flux to a (usually cylindrical/spherical) volume integral.06Stokes' Theorem
Section 1 · L11. Stokes' theorem ∬ (∇×F)·n̂ dS = ∮ F·dr: open surface, boundary curve, right-hand-rule orientation, and the link back to conservative fields.07Systems of First-Order ODEs: Eigenvalue Solution
Section 2 · L12. Solving ẋ = Ax via eigenvalues/eigenvectors — distinct-real, repeated/deficient (generalised eigenvector), and complex-conjugate cases using the provided general-solution forms.08Phase Portraits and Critical-Point Classification
Section 2 · L13–L16. Classifying critical points by eigenvalues (node/saddle/star/spiral/centre, stable vs unstable), sketching phase portraits, and linearising non-linear systems via the Jacobian.09Laplace Transforms
Section 3 · L17–L21. Transform definition and table, transforms of derivatives, solving IVPs and systems, s- and t-shifting, unit step and Dirac delta, and the convolution theorem for integral equations.10Sequences and Series: the Convergence Toolkit
Section 4 · L22–L26. Partial sums, geometric and p-series, the full battery of convergence tests, absolute vs conditional convergence, power-series radius/interval, and Taylor polynomials and series.11Fourier Series and Fourier Integrals
Section 5 · L27–L30. Fourier series (Euler formulae, ω=π/L), Parseval/energy, cosine/sine series, odd/even periodic extensions and midpoint convergence, periodic forcing, and Fourier integrals.12Second-Order PDEs and Separation of Variables
Section 6 · L31–L35. Classifying 2nd-order PDEs by B²−4AC (Laplace/wave/diffusion) and separation of variables — separation constant, eigenvalue cases, BCs, superposition, coefficient matching.It assembles six near-independent technique families: vector calculus (multiple integrals, coordinate changes, line and surface integrals, and Gauss' and Stokes' theorems), systems of first-order ODEs with phase-plane analysis and linearisation, Laplace transforms with shift theorems and convolution, sequences and series with the full convergence toolkit and Taylor series, Fourier series and integrals, and second-order PDEs by separation of variables.
The final exam is 65% of your grade — 3 hours, in person, closed-book, with no calculator and no notes — but you are given a 5-page formula sheet, so the marks come not from formula recall but from recognising which technique a question wants, executing it accurately by hand, and justifying it (name the theorem, check its conditions, show full working, use correct notation).
There is also a genuine in-semester hurdle: the mid-semester test, quizzes and assignments together are worth 35%, and you must score at least 17.5% of that combined 35% to pass the subject at all — so you can ace the exam and still fail if you skip the in-semester work. This guide teaches each of the six families to exam standard, mapping every chapter to the past-paper question it recurs in.
How MAST20029 is assessed
| Component | Weight | Format |
|---|---|---|
| Final exam | 65% | End of semester · in person · 15 min reading + 3 hours writing · closed-book, no calculator, no notes/cheat sheets, no MATLAB · 5-page formula sheet provided · 11 questions, 120 marks (6–16 each), roughly in lecture order · marks for correct method, full working, accuracy and notation (exam date subject to confirmation in your personal timetable) |
| Mid-semester test · hurdle | 15% | In person during allocated lecture time · 45 minutes · closed-book, no calculator · covers Lectures 1–14 (to end of Week 5) + practice sheets 2–6 · 40 marks · relevant section of the formula sheet provided · counts toward the in-semester hurdle (date subject to confirmation) |
| Assignments (3 × 5%) · hurdle | 15% | Three written assignments due across the semester · handwritten, single PDF, submitted in Canvas · counts toward the in-semester hurdle (dates subject to confirmation) |
| Quizzes (best 5 of 6) · hurdle | 5% | Six fortnightly online multiple-choice quizzes, 30 minutes each once started · 1% each, best 5 of 6 count (one free miss) · no extensions or special consideration · counts toward the in-semester hurdle |
Gauss' divergence theorem — flux out of a closed cylinder
- 2 marksRecognise a CLOSED surface with outward normal, so convert the flux to a volume integral by Gauss' theorem rather than integrating over three faces separately. Compute the divergence: ∇·F = ∂(x³)/∂x + ∂(y³)/∂y + ∂(z³)/∂z = 3x² + 3y² + 3z².
- 2 marksApply Gauss: flux = ∭V 3(x² + y² + z²) dV. The region is a cylinder, so switch to cylindrical coordinates x²+y² = r², dV = r dr dθ dz, with 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ 2.
- 2 marksIntegrate inner (z): ∫₀² (r² + z²) dz = 2r² + 8/3. Then middle (r), carrying the Jacobian r: ∫₀² (2r² + 8/3)·r dr = ∫₀² (2r³ + (8/3)r) dr = 8 + 16/3 = 40/3.
- 2 marksIntegrate outer (θ) and restore the factor 3: flux = 3 · ∫₀²π dθ · (40/3) = 3 · 2π · 40/3 = 80π.
Key terms
- Divergence and curl
- For a vector field F = F₁ i + F₂ j + F₃ k, the divergence ∇·F = ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z is a SCALAR (net outflow per unit volume); the curl ∇×F is a VECTOR (local rotation), computed from the i,j,k determinant. Two identities recur: curl(grad φ)=0 and div(curl F)=0.
- Jacobian
- The factor |J| = |∂(x,y)/∂(u,v)| that corrects the area/volume element under a change of variables: dA = |J| du dv. In this course polar and cylindrical give J=r, and spherical (with x=r cos θ sin φ) gives J=r² sin φ. Forgetting the Jacobian is the single most common multiple-integral error.
- Conservative field
- A field F with curl F = 0 on a simply connected domain, equivalently F = ∇φ for a potential φ. Then a line integral depends only on the endpoints, ∫_C F·dr = φ(end) − φ(start), and the work around any closed loop is 0 — so you never grind the integral when you can spot conservativity.
- Critical point classification
- For the linear system ẋ = Ax, the type of the origin is read straight from the eigenvalues of A: two negative reals → stable node, opposite signs → saddle, complex with negative real part → stable spiral, pure imaginary → centre (and the unstable mirrors). The provided formula-sheet table is your decision rule.
- Convolution theorem
- The Laplace identity L{(f∗g)(t)} = F(s)G(s), where (f∗g)(t) = ∫₀ᵗ f(τ)g(t−τ) dτ. It turns an integral equation (an unknown y under ∫₀ᵗ y(τ)g(t−τ) dτ) into ordinary algebra in F(s) — the exam expects you to NAME the theorem you used.
MAST20029 FAQ
Is MAST20029 hard?
It is demanding because of its breadth, not because any one idea is exotic: six near-independent technique families (vector calculus, ODE systems, Laplace transforms, series, Fourier, PDEs) are examined across an 11-question, 120-mark closed-book paper in 3 hours.
The good news is that the exam is method-first — the formula sheet is provided, so what separates marks is recognising which tool a question wants and executing it cleanly by hand. Students who drill the signature question for each section and practise full, justified working find it very tractable.
Is there a hurdle in MAST20029?
Yes — a real one. The source states you must obtain at least 17.5% out of the combined 35% for the mid-semester test, quizzes and assignments to pass the subject, regardless of your exam mark. In other words you cannot coast on the final and skip the in-semester work: keep up with the three assignments and the fortnightly quizzes (best 5 of 6 count, so you get one free miss) and treat the mid-semester test seriously.
Do I need to memorise the formulas?
No. A 5-page formula sheet is provided in both the final exam and the relevant section in the mid-semester test, and it gives the transforms, integrals, the critical-point classification table, standard limits and the trig/hyperbolic identities. The marks come from method selection, justification, correct notation and full working — so practise choosing and applying the right tool, not rote recall.
Knowing WHERE each formula sits on the sheet saves time under pressure.
Can I use a calculator or MATLAB in the exam?
No. The final exam and the mid-semester test are both closed-book with no calculator, no notes or cheat sheets and no MATLAB or symbolic software. MATLAB is used in the subject for learning but is explicitly NOT examined. Everything in the exam is done by hand using the provided formula sheet, so rehearse exact surd and fraction arithmetic (for example leaving an answer as (4/15)(9√3 − 8√2 + 1) rather than a decimal).
What does the mid-semester test cover versus the final?
The mid-semester test (45 minutes, 15%) covers Lectures 1–14 (to the end of Week 5) plus practice sheets 2–6 — that is the early vector-calculus material and the start of ODE systems.
The final exam (3 hours, 65%) covers all lecture, tutorial and problem-sheet material across the six sections except MATLAB, with 11 questions arranged roughly in lecture-topic order (vector calculus, ODE systems and phase portraits, Laplace, series, Fourier, then PDEs).
How to study for the exam
Treat MAST20029 as six separate skills that share one exam habit, not one giant subject. (1) Clear the hurdle first: the assignments, fortnightly quizzes and mid-semester test are 35% combined and you must score at least 17.5% of that to pass at all, so never let the in-semester work slide while chasing exam practice.
(2) Build a decision tree per section — for an integral, ask which coordinate system the region suggests (polar/cylindrical/spherical) and never drop the Jacobian; for a series, run the divergence test first then pick comparison/ratio/root/integral/Leibniz by the SHAPE of aₙ; for an ODE system, go straight to the eigenvalues.
(3) Learn the formula sheet's geography — you are given every transform, the critical-point table and the standard limits, so spend exam time on setup and justification, not searching. (4) Rehearse the justified-working ritual the markers reward: name the theorem or test, check its conditions, show every substituted line, and state which theorem you used (explicitly required on Laplace and series questions).
(5) Work the past papers by section — the 2024 and 2025 exams map almost 1:1 to the six sections, so the signature question for each recurs nearly verbatim. (6) Practise exact arithmetic by hand (surds, fractions, factorials) because there is no calculator — a clean (1/2)ln 3 beats a wrong decimal.
