MTH2010 Multivariable Calculus
The Complete Exam Bible · S2 2026MTH2010 Overview
- Monash University
- Level 2 mathematics
- Semester 2, 2026
- Seminars and applied classes
- Moodle unit materials
MTH2010 Multivariable Calculus is a Monash University mathematics unit that carries calculus from one variable to several.
- Assessed by Quizzes and two in-class tests, then a final examination
- Hardest step Setting up limits, regions and orientations correctly
- How to prepare Rework examples, then sit a practice final under time
How MTH2010 is assessed
| Component | Weight | Format |
|---|---|---|
| Applied class quizzes | 10% | Eleven quizzes of 1% each at the end of applied classes, best ten counted; attend in person to receive credit |
| Seminar quizzes | 10% | Five short quizzes of 2% each held during seminars |
| Mid-semester test I | 15% | Week 4, in class; practice papers set 60 minutes with no outside materials or devices |
| Mid-semester test II | 15% | Week 9, in class |
| Final examination | 50% | Final exam period; two practice finals and a final exam formula sheet are posted on Moodle |
What MTH2010 covers
Follow the unit week by week, from lines, planes and limits through optimisation and multiple integrals to line integrals, surface integrals and the theorems of Green, Stokes and Gauss.
Vectors, Lines, Planes and Curves in Space
Week 1. Dot and cross products, parametric and symmetric lines, planes from a point and a normal, parametric curves, tangent vectors and tangent lines.Week 1 of the unit sets up the geometry that every later topic relies on. A point in three-dimensional space is identified with its position vector, and lines, planes and curves are all described by a base point together with direction vectors.
- Dot product: angle between vectors and the test for orthogonality
- Cross product: a vector perpendicular to two given directions, and parallelepiped volume
- Lines: parametric form through two points and symmetric form with zero components omitted
- Planes: point and normal form, parametric form, and conversion between them
- Line against plane: direction dotted with the normal, then one point substituted
- Parametric curves: componentwise derivatives and the regularity assumption
Functions of Several Variables, Limits and Continuity
Weeks 1 and 2. Domain and range, level curves and level surfaces, two-variable limits, path tests, the squeeze theorem and continuity.This chapter covers the shift from functions of one variable to functions of two or three. A function of two variables has a domain in the plane, a range of output values and a graph that is a surface in space.
- Domain and range: inequalities for logarithms and roots, then range from level sets
- Level curves and contour maps: every case of f = c, including a point or the empty set
- Graphs versus level sets: why spheres need a level-set description
- Two-variable limits: the definition requires agreement along every path
- Disproving a limit: two paths, or a curved path when lines all agree
- Proving a limit: limit laws, the squeeze theorem, or epsilon-delta
Partial Derivatives, Tangent Planes and Linear Approximation
Weeks 2 and 3. Partial derivatives from the limit definition, higher partials and Clairaut, tangent planes, linearisation, differentiability and PDEs.Partial derivatives measure how a function of several variables changes when only one input moves.
- Partial derivatives: hold the other variables fixed and differentiate
- Limit definition: required at seams of piecewise functions
- Higher partials: the order convention for mixed partials and Clairaut's theorem
- Tangent plane: built from the two slice tangent vectors and a normal
- Linearisation: value plus partials times displacements, used for estimates
- Differentiability: relative error tends to zero; continuous partials suffice
Chain Rule, Implicit Differentiation and the Gradient
Week 4. Multivariable chain rule, implicit function theorem, implicit derivatives, directional derivatives, the gradient and normals to level sets.Week 4 extends differentiation to compositions and to functions defined implicitly, then introduces the gradient, the single most useful vector in the unit.
- Chain rule: one product per route in the dependency tree
- Evaluation points: outer partials at the inner point, inner partials at (s, t)
- Total versus partial derivatives: the homework's fake proof
- Implicit function theorem: three conditions to check before using the formula
- Implicit derivatives: dy/dx = -F_x / F_y and z_x = -F_x / F_z
- Directional derivatives: unit vector dotted with the gradient
Maxima, Minima and Lagrange Multipliers
Week 5. Critical points, the second derivatives test, absolute extrema on closed bounded regions, and constrained optimisation with Lagrange multipliers.Optimisation in two variables is the subject of Week 5 and a frequent source of mid-semester and final examination questions.
- Critical points: zero gradient or no gradient
- Second derivatives test: the sign of AC - B² and of A
- Saddle points: why they are never extrema and can be discarded
- Extreme value theorem: closed, bounded and continuous
- Absolute extrema routine: interior, edges, corners, comparison
- Lagrange multipliers: parallel gradients plus the constraint
Double Integrals and Fubini's Theorem
Week 6. Riemann sums, iterated integrals, type I and type II regions, reversing the order of integration and bounding integrals.Week 6 defines the double integral and turns it into something computable.
- Riemann sums: the definition of the double integral
- Volume and area: the two geometric readings of the integral
- Fubini on rectangles: either order for continuous integrands
- Type I and type II regions: limits written as inequalities
- Reversing the order: sketch, redescribe, re-integrate
- Splitting regions: additivity over pieces with disjoint interiors
Polar Coordinates and Applications of Double Integrals
Week 7. Polar regions and the area element r dr dθ, mass and charge, moments, centre of mass and joint probability densities.Polar coordinates make discs, annuli and sectors as easy to integrate over as rectangles. With x = r cos(θ) and y = r sin(θ), a region bounded by circles about the origin and rays from it becomes a rectangle in the r-theta plane.
- Polar substitution: x = r cos(θ), y = r sin(θ)
- Area element: dA = r dr dθ and why the r appears
- Polar rectangles: exact area and separation into products
- When to use polar: simpler region or simpler integrand
- Converting boundaries: circles through the origin and rays
- Mass and charge: integrating a density over the region
Triple Integrals in Cylindrical and Spherical Coordinates
Week 8. Triple integrals over solids, region types, cylindrical and spherical coordinates, volume elements, volume and average value.Week 8 moves from regions in the plane to solids in space. A triple integral is again the limit of Riemann sums, now over small boxes, and integrating 1 gives the volume of the solid.
- Triple integrals as limits of sums over small boxes
- Region types I, II and III and the inner integral
- Volume as the integral of one, and average value
- Cylindrical coordinates: dV = r dr dθ dz
- Spherical coordinates: rho, φ and θ and their ranges
- Spherical volume element: ρ² sin(φ)
Change of Variables, Vector Fields and Line Integrals
Week 9. Jacobians and general changes of variables, vector fields, scalar line integrals with arc length, and work and circulation integrals.Week 9 opens the vector calculus part of the unit with three topics listed on the weekly page: change of variables in multiple integrals, vector fields, and integrals along curves. A change of variables replaces x and y by functions of new variables u and v.
- Jacobian determinant: the local area scale factor of a substitution
- Polar coordinates as a change of variables with Jacobian r
- Choosing substitutions from the shape of the region
- Vector fields: pictures, flow and sign predictions
- Scalar line integrals: ds = |r'(t)| dt and arc length
- Vector line integrals: F(r(t)) dotted with r'(t)
Conservative Fields and Green's Theorem
Week 10. Gradient fields and potentials, the fundamental theorem for line integrals, path independence, and Green's theorem with orientation.Week 10 lists four topics: conservative vector fields, the fundamental theorem for line integrals, path independence, and Green's theorem. A vector field is conservative when it is the gradient of a scalar function, called a potential.
- Conservative fields and potentials
- Fundamental theorem: potential at the end minus potential at the start
- Path independence and zero circulation around closed curves
- Component test P_y = Q_x and the no-holes hypothesis
- Building a potential by integrating and matching
- Green's theorem: boundary circulation equals an area integral
Parametric Surfaces and Surface Integrals
Week 11. Parametric surfaces, tangent planes and normals, the surface element, surface area, oriented surfaces and flux integrals.Week 11 develops surfaces the way Week 1 developed curves. A parametric surface is the image of a function r(u, v) of two parameters.
- Parametric surfaces and grid curves
- Tangent vectors r_u and r_v and the tangent plane
- Normal vector as the cross product r_u x r_v
- Surface element dS and surface area
- Graphs as surfaces: sqrt(1 + g_x² + g_y²) dA
- Scalar surface integrals: mass of a shell
Stokes' Theorem and the Divergence Theorem
Week 12. Curl and divergence, Stokes' theorem with the right-hand rule, the divergence theorem for closed surfaces, and applications.The final week of the unit lists Stokes' theorem, the divergence theorem and applications.
- Curl and divergence of a vector field
- Stokes' theorem: circulation equals flux of the curl
- Right-hand rule for matching orientations
- Replacing a surface by a simpler one with the same boundary
- Divergence theorem: outward flux equals the volume integral of divergence
- Closing an open surface with a lid
In Semester 2, 2026 it runs as weekly seminars plus a two-hour applied class running from Week 2 to Week 12, where students work in small groups at the whiteboard on problems from the previous week's seminar. The unit shares its Moodle site with the honours unit MTH2015, which adds advanced topics and its own assessment; this guide follows MTH2010.
The first half is differential calculus in several variables.
It begins with lines, planes and parametric curves in space, then level curves, two-variable limits and continuity, partial derivatives, tangent planes and linear approximation.
The chain rule, implicit differentiation and the gradient lead to optimisation: classifying critical points with the second derivatives test, finding absolute extrema on closed and bounded regions, and constrained optimisation with Lagrange multipliers.
The second half is integration.
Double integrals are set up as iterated integrals over type I and type II regions, then converted to polar coordinates for mass, centre of mass and probability. Triple integrals follow in cylindrical and spherical coordinates.
The last four weeks build vector calculus: changes of variables, vector fields, line integrals, conservative fields and Green's theorem, parametric surfaces and flux, and finally Stokes' theorem and the divergence theorem.
The stated learning outcomes include explaining the derivative as a tangent plane together with the gradient and directional derivative, finding and classifying extrema including constrained ones, computing volume, surface and line integrals using Cartesian, cylindrical or spherical coordinates, using the theorems of Green, Stokes and Gauss, and communicating mathematical reasoning in writing.
The primary texts are the open textbooks Multivariable Calculus and Vector Calculus by Feldman, Rechnitzer and Yeager.
Worked example · free
Classify the critical points of a cubic surface
- 1Set the partial derivatives to zero: f_x = 3x² - 3 = 0 gives x = 1 or x = -1, and f_y = 2y - 4 = 0 gives y = 2.
- 1The critical points are (1, 2) and (-1, 2).
- 1Second partials: f_xx = 6x, f_xy = 0 and f_yy = 2, so D = f_xx f_yy - f_xy² = 12x.
- 1At (1, 2): D = 12 and f_xx = 6 are both positive, so a local minimum with f(1, 2) = 1 - 3 + 4 - 8 = -6.
- 1At (-1, 2): D = -12 is negative, so a saddle point, where f(-1, 2) = -1 + 3 + 4 - 8 = -2.
Key terms
- Level curve
- The set of points in the domain of a function of two variables where the function takes one fixed value c.
- Partial derivative
- The rate of change of a function in one coordinate direction while every other variable is held fixed.
- Linearisation
- The first-order approximation built from the value and both partial derivatives at a point; its graph is the tangent plane.
- Gradient
- The vector of partial derivatives, which points in the direction of fastest increase and is normal to level sets.
- Directional derivative
- The rate of change of a function along a unit vector, equal to that unit vector dotted with the gradient.
- Critical point
- A point where the gradient is zero or fails to exist; the only candidates for interior local extrema.
- Saddle point
- A critical point that is neither a local maximum nor a local minimum because the function rises in some directions and falls in others.
- Lagrange multiplier
- The scalar that makes the gradient of the objective a multiple of the gradient of the constraint at a constrained extremum.
- Jacobian
- The determinant of partial derivatives of a change of variables, whose absolute value scales area or volume.
- Conservative field
- A vector field that is the gradient of a potential, so its line integrals depend only on the endpoints.
- Flux integral
- The integral of the normal component of a vector field over an oriented surface.
- Divergence
- The sum of the partial derivatives of the components of a vector field, measuring net outflow per unit volume.
MTH2010 FAQ
How is the unit assessed?
The published summary lists applied class quizzes worth 10%, seminar quizzes worth 10%, two mid-semester tests worth 15% each and a final examination worth 50%. The Handbook describes the same split as Demonstration 50% and Examination 50%.
What do we know about the final examination?
It carries half of the unit mark and is held in the final exam period. Two practice final examinations with solutions and a final exam formula sheet are released in the Week 11 section of Moodle; check the exam notice for timing and permitted materials.
When are the mid-semester tests and what are they like?
The assessment summary places test I in Week 4 and test II in Week 9, both in class. The practice papers allow sixty minutes, contain several multi-part problems and permit no outside materials or devices.
How do the applied class quizzes count?
There are eleven quizzes worth one percent each, and the best ten count towards the final grade. Each follows an applied class, and credit requires attending that class in person, so plan attendance as part of revision.
Which textbooks does the unit use?
The primary resources are the open-source textbooks Multivariable Calculus and Vector Calculus by Feldman, Rechnitzer and Yeager. Each weekly page lists the matching sections, and Stewart's Calculus is suggested as an additional reference.
Which topics cause the most trouble?
Two-variable limits, where one path can disprove but never prove a limit, and the setup of multiple integrals, where a missing area or volume factor or a reversed orientation changes the answer. Both reward a clear first line.
Is a formula sheet provided?
Yes for the final. A final exam formula sheet is posted in the Week 11 section from 12 October 2026. Annotate a copy with where each formula is used so that the sheet guides method choice, not just recall.
Questions students ask in each MTH2010 chapter
- How do I find a plane through three points?
- Form two edge vectors from one of the points, take their cross product to get a normal, then write the point-normal equation. Check the answer by substituting all three original points. Chapter 01 · Vectors, Lines, Planes and Curves in Space
- Why are straight-line paths not enough to show a limit exists?
- Some functions agree along every straight line through a point yet behave differently along a curve, such as a parabola. Agreement along any collection of paths is never a proof; a bound valid for all nearby points is needed. Chapter 02 · Functions of Several Variables, Limits and Continuity
- When must I use the limit definition of a partial derivative?
- Use it where the derivative rules do not apply, for example at the join between two pieces of a piecewise function or where a fractional power is not differentiable. Write the difference quotient explicitly and evaluate the limit, including one-sided limits if needed. Chapter 03 · Partial Derivatives, Tangent Planes and Linear Approximation
- Where do I evaluate the partial derivatives in the chain rule?
- Evaluate the outer function's partials at the inner point, meaning the values of x and y produced by the given s and t. Evaluate the inner derivatives at s and t themselves, then multiply along each route and add. Chapter 04 · Chain Rule, Implicit Differentiation and the Gradient
- What counts as a critical point in this unit?
- A point where both partial derivatives are zero, or where at least one of them fails to exist. The second case matters for functions such as cones and absolute values, whose extremes sit at sharp points. Chapter 05 · Maxima, Minima and Lagrange Multipliers
- When can I change the order of integration?
- When the integrand is continuous and the region can be described both ways, Fubini's theorem makes the two iterated integrals equal. Redraw the region from the given limits and rewrite them for the other order. Chapter 06 · Double Integrals and Fubini's Theorem
- Why does polar integration need the extra factor r?
- A small polar box between radii r and r + dr and angles θ and θ + dθ has sides dr and r dθ, so its area is r dr dθ. Without the r, areas and every integral built on them come out wrong. Chapter 07 · Polar Coordinates and Applications of Double Integrals
- How do I choose between cylindrical and spherical coordinates?
- Use cylindrical coordinates for solids with an axis of symmetry, such as cylinders, paraboloids and cones described by z and r. Use spherical coordinates when spheres centred at the origin or cones about the z-axis bound the solid. Chapter 08 · Triple Integrals in Cylindrical and Spherical Coordinates
- What does the Jacobian measure in a change of variables?
- Its absolute value is the factor by which small areas are stretched when moving from the u-v plane to the x-y plane, so it multiplies the integrand after the substitution. Chapter 09 · Change of Variables, Vector Fields and Line Integrals
- How do I test whether a plane vector field is conservative?
- Check that P_y equals Q_x and that the field's domain has no holes. If both hold, a potential exists; build it by integrating P in x and matching the y-derivative with Q. Chapter 10 · Conservative Fields and Green's Theorem
- How do I find a normal vector to a parametric surface?
- Differentiate r(u, v) with respect to each parameter to get two tangent vectors, then take their cross product. Reverse the order of the cross product if the requested orientation points the other way. Chapter 11 · Parametric Surfaces and Surface Integrals
- How do I match orientations in Stokes' theorem?
- Use the right-hand rule. Point your thumb along the chosen normal of the surface; your fingers then curl in the direction the boundary must be traversed. Reversing either choice reverses the sign. Chapter 12 · Stokes' Theorem and the Divergence Theorem
How to study for the exam
Work in the order of the semester, but revise in three blocks: differential calculus (Chapters 1 to 5), multiple integrals (Chapters 6 to 8) and vector calculus (Chapters 9 to 12). For each chapter, rework the guide's example without looking, then attempt the applied class sheet for that week and the matching homework sheet, which the unit describes as harder than the applied problems.
Keep one page of setups rather than answers: the region inequalities for each integral, the volume element for each coordinate system, and the orientation rule for each theorem. Two weeks before the exam, sit one practice final under timed conditions, mark it from the solutions, and spend the following days on the questions where the setup, not the arithmetic, went wrong.
Use the formula sheet during practice so that you know where everything on it is and when it applies.
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