MAST20029: pass the exams, not just read the notes
Your complete guide to University of Melbourne's engineering mathematics unit. See where the marks are, work real practice questions, and study with an AI tutor that knows MAST20029.
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Worked example
Solve the first-order linear differential equation dy/dx = 3y with initial condition y(0) = 2. What is y(x)?
dy/dx = 3y is separable: divide by y and integrate, giving (1/y) dy = 3 dx.
Apply y(0) = 2: A e^0 = A = 2.
So y = 2 e^(3x).
The trap: Treating dy/dx = 3y as if it grew linearly (y = 2 + 3x). A rate proportional to the quantity itself gives exponential, not linear, growth; the solution is 2e^(3x). classic slip!
One exam decides 65% of your grade. This whole page is built around that.
Overview
What MAST20029 is, and where it sits
MAST20029 Engineering Mathematics is the University of Melbourne's second-year mathematics subject for engineering and science students, taught in the School of Mathematics and Statistics. It builds the applied-mathematics toolkit engineering subjects assume: multivariable and vector calculus (partial derivatives, gradient, divergence and curl, multiple and line integrals), ordinary differential equations, Laplace transforms, and systems of linear differential equations with phase-plane analysis.
The emphasis is fluent, accurate technique applied to problems, assessed largely under exam conditions: a 65% final and a 15% mid-semester test dominate the grade, with small assignments and quizzes keeping practice continuous. The subject rewards students who drill the methods until they are automatic and can move confidently between the calculus, differential-equations and linear-systems toolkits.
Official outline: handbook.unimelb.edu.au · MAST20029 outline. Always treat the official outline and the exam timetable as authoritative.
Difficulty & time commitment
Is MAST20029 hard, and how much time does it take?
MAST20029 is manageable if you keep a weekly rhythm and treat the back half as the main event. The pattern is consistent: it starts gently and steepens, and the heaviest assessment is the part that separates grades.
The difficulty curve and the assessment weighting point the same way: the back half is harder and worth more. Front-loading effort there is the highest-return decision in the unit.
Is this unit for you
Who tends to do well, and who tends to struggle
You will likely do well if
- You have solid first-year calculus and linear algebra and drill the new methods until they are automatic.
- You practise by hand and timed, since the 65% final rewards fast, accurate technique.
- You keep pace through the differential-equations and Laplace block, the steepest part of the subject.
You may struggle if
- You fall behind in vector calculus; the later ODE and linear-systems material assumes it.
- You rely on software for steps the exam expects you to do by hand.
- You under-practise the heavily weighted final, treating the small assignments as the main event.
- Build a methods sheet: gradient/divergence/curl, line and multiple integrals, ODE solution types, Laplace transform pairs.
- Work past exams by hand and timed, prioritising the differential-equations and Laplace questions.
- Practise phase-plane analysis of linear systems until classifying critical points is automatic.
Syllabus
The 12 topics, topic by topic
The exam-weight marker on each topic shows where the marks concentrate. The amber topics carry the highest exam weight.
T1 · 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.
T2 · Double 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.
T3 · Polar, 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.
T4 · Line, 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.
T5 · Gauss' 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.
T6 · Stokes' 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.
T7 · Systems 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.
T8 · Phase 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.
T9 · Laplace 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.
T10 · Sequences 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.
T11 · Fourier 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.
T12 · Second-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.
How it's assessed
Assessment structure
| Component | Weight | Format & timing |
|---|---|---|
| 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 | 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). YES. |
| Assignments (3 × 5%) | 15% | Three written assignments due across the semester · handwritten, single PDF, submitted in Canvas · counts toward the in-semester hurdle (dates subject to confirmation). YES. |
| Quizzes (best 5 of 6) | 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. YES. |
- Pass on a weighted average of at least 50%. No single-component hurdle unless noted; confirm against the official subject page.
This is an exam-cram unit. With the exams at 80% of the grade and the final exam alone at 65%, your result is overwhelmingly decided by how well you perform under time pressure.
How to actually pass it
A weekly rhythm, two checklists, and the traps to avoid
The unit rewards consistency over cramming, and practice over re-reading. Here is the loop that works, then what to have nailed before each exam.
The weekly loop
Before the mid-semester checklist
Before the final heaviest topics
- Prioritise ODEs, Laplace transforms and linear systems, the heavily weighted back half.
- Rehearse vector-calculus operations (gradient, divergence, curl, line and surface integrals).
- Drill Laplace-transform and inverse-transform problems until the table is automatic.
- Work every past final by hand end-to-end under time pressure.
The mistakes that cost marks
Linearising a proportional rate. Equations like dy/dx = ky give exponential solutions; treating them as linear growth is a classic differential-equations error.
Falling behind in vector calculus. The ODE, Laplace and linear-systems material assumes fluent multivariable and vector calculus; an early gap compounds through the steep second half.
Under-practising the final. At 65% the final dominates. Timed hand practice on the hardest topics is where the grade is won or lost.
Teaching team
Who teaches MAST20029
The bios below are factual. We do not rate lecturers; any star ratings are submitted by students who have taken MAST20029.
Associate Professor Marcus Brazil
Subject coordinator and lecturer (early stream) for MAST20029 in the School of Mathematics and Statistics, University of Melbourne.
Associate Professor Leo Tzou
Lecturer (late stream) for MAST20029 in the School of Mathematics and Statistics, University of Melbourne.
Dr Khurram Kamran
Tutor coordinator for MAST20029 in the School of Mathematics and Statistics, University of Melbourne.
Teaching team as listed in the unit materials reviewed. AskSia does not rate lecturers; star ratings are submitted by students who have taken MAST20029.
Formula & concept sheet
The vocabulary and formulas you must own
- Gradient
- grad f = (df/dx, df/dy, df/dz): the vector of partial derivatives pointing in the direction of steepest increase of a scalar field.
- Divergence and curl
- Divergence measures the net outflow of a vector field at a point; curl measures its rotation. Both are built from partial derivatives of the field components.
- Separable ODE
- An equation dy/dx = g(x)h(y) solved by separating variables and integrating both sides; equations of the form dy/dx = ky give exponential solutions.
- Laplace transform
- L{f(t)} = integral of e^(-st) f(t) dt: converts a differential equation in t into an algebraic equation in s, solved and then inverted.
- Linear system / phase plane
- A system dx/dt = Ax analysed via the eigenvalues of A; the signs and types of eigenvalues classify the critical point (node, saddle, spiral, centre).
Common acronyms: ODE · PDE · div · curl · grad · MST.
Where it fits
Prerequisites, related units & why it matters
Second-year subject building on first-year calculus (e.g. Calculus 2) and linear algebra; prerequisite-gated. Check the UniMelb Handbook.
Your MAST20029 study toolkit
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FAQ
Frequently asked questions
Is MAST20029 (Engineering Mathematics) hard?
It is moderate-to-hard: technical and exam-heavy, with a 65% final and dense content across vector calculus, differential equations, Laplace transforms and linear systems. Strong first-year maths and consistent hand practice make it manageable.
How is MAST20029 assessed?
A 65% final exam, a 15% mid-semester test, three assignments worth 5% each, and quizzes worth 5% (best five of six). The final dominates the grade.
What maths does it cover?
Multivariable and vector calculus (gradient, divergence, curl, multiple and line integrals), ordinary differential equations, Laplace transforms, and systems of linear differential equations with phase-plane analysis.
What background do I need?
First-year calculus (such as Calculus 2) and linear algebra. The subject assumes that toolkit and extends it to applied engineering methods.
What is the hardest part?
The differential-equations, Laplace-transform and linear-systems block in the second half, which is both the most technical material and the most heavily examined.
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