MAST20029 Engineering Mathematics

How MAST20029 is assessed

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.

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.

Worked example · free

Gauss' divergence theorem — flux out of a closed cylinder

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

Questions students ask in each MAST20029 chapter

What is the difference between divergence and curl?

Divergence is a scalar that measures how much the field spreads out from (or into) a point — like the net flow out of a tiny box. Curl is a vector that measures how much the field circulates around a point — like the rotation a tiny paddle wheel would feel. A field can have one without the other: a pure source has divergence but no curl, while a pure swirl has curl but no divergence. Chapter 01 · Vector Fields, Divergence and Curl

When should I change the order of integration?

Two triggers. First, when the inner integral has no elementary antiderivative in the given order (the classic examples are e^(x²), e^(x³), sin(x²), and (sin x)/x) but becomes a simple substitution after swapping. Second, when one order forces a region to be split into several pieces but the other order covers it in one. Sketch the region, re-describe it, then swap. Chapter 02 · Double Integrals and Change of Order

How do I know which coordinate system to use?

Let the region's symmetry decide. A circular or annular boundary, or an integrand built from x²+y², points to polar; a cylinder or an axis of symmetry points to cylindrical; a ball, sphere or an x²+y²+z² integrand points to spherical. The right system turns curved limits into constant ones and usually simplifies the integrand. Chapter 03 · Polar, Cylindrical and Spherical Coordinates

How do I test whether a field is conservative?

Compute the curl: on a simply connected domain, F is conservative if and only if curl F = 0. In two dimensions this reduces to ∂F₂/∂x = ∂F₁/∂y. If the curl is zero you can find a potential φ with F = ∇φ; if it is non-zero the field is not conservative and you must integrate directly. Chapter 04 · Line, Work and Surface Integrals; Conservative Fields

When do I use Gauss' theorem instead of integrating directly?

Whenever you need the flux through a closed surface and computing the divergence is easier than integrating over each face. The theorem turns a multi-face surface integral into one volume integral of ∇·F, which is usually far quicker — especially when the divergence is simple or constant. Chapter 05 · Gauss' Divergence Theorem

What is the difference between Stokes' theorem and Gauss' theorem?

Gauss relates a CLOSED surface's flux to a volume integral of the divergence; Stokes relates an OPEN surface's curl flux to a line integral around its boundary. Gauss needs no boundary curve (closed surface); Stokes needs one (open surface). Read whether the surface is closed to choose the right theorem. Chapter 06 · Stokes' Theorem

How do I solve a linear system ẋ = Ax?

Find the eigenvalues from det(A − λI) = 0, then the eigenvectors from (A − λI)w = 0, and assemble the general solution using the case that matches the eigenvalues: distinct real, repeated/deficient, or complex. The formula sheet gives the general-solution form for each case, so the work is in computing eigenvalues and eigenvectors correctly. Chapter 07 · Systems of First-Order ODEs: Eigenvalue Solution

How do I classify a critical point from the eigenvalues?

Two real same-sign eigenvalues give a node (stable if both negative, unstable if both positive); real opposite signs give a saddle (always unstable); complex with nonzero real part give a spiral (stable if the real part is negative); pure imaginary give a centre. Repeated eigenvalues give star or improper nodes depending on the geometric multiplicity. The formula-sheet table is the decision rule. Chapter 08 · Phase Portraits and Critical-Point Classification

How does the Laplace method solve an ODE?

Transform both sides of the IVP, which turns derivatives into algebra and folds in the initial conditions via L{f'} = sF − f(0) and L{f''} = s²F − sf(0) − f'(0). Solve the resulting algebraic equation for F(s), then invert using the provided table, partial fractions, and the shift theorems to recover f(t). Chapter 09 · Laplace Transforms

Which convergence test should I use?

Always run the divergence test first — if the terms do not go to zero, you are done. Then choose by the shape of aₙ: factorials or nth powers suggest the ratio or root test; a rational or bounded-over-power form suggests comparison or limit comparison with a p-series; a positive decreasing term you can integrate suggests the integral test; an alternating sign suggests Leibniz. Match the tool to the structure. Chapter 10 · Sequences and Series: the Convergence Toolkit

When do I get a cosine series versus a sine series?

Symmetry decides. An even function (or an even periodic extension) has only cosine terms — a₀ and aₙ — because the sine integrals vanish. An odd function (or odd extension) has only sine terms bₙ. For a function given on (0, L) you CHOOSE the extension: odd for a sine series, even for a cosine series, which halves the work. Chapter 11 · Fourier Series and Fourier Integrals

How do I classify a second-order PDE?

Compute B² − 4AC from the coefficients of u_xx, u_xy and u_yy. If it is negative the PDE is elliptic (like Laplace's equation, steady states); if zero it is parabolic (like the diffusion/heat equation); if positive it is hyperbolic (like the wave equation). The class tells you the physical character and the appropriate boundary/initial data. Chapter 12 · Second-Order PDEs and Separation of Variables

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.