UNSW MATH2011 Several Variable Calculus: Study Guide, Practice

MATH2011 Several Variable Calculus is UNSW Sydney's second-year course in multivariable calculus, taken after Mathematics 1B by students in mathematics, physics, science and other quantitative programs. It is worth 6 units of credit and runs in Term 1. This guide maps every MATH2011 topic to free practice, so you can build each skill by solving problems with instant feedback.

What MATH2011 covers

The UNSW Handbook describes MATH2011 as an introduction to differential, integral and vector calculus for functions of more than one variable. The listed topics fall into three blocks:

1. Differential calculus: functions of several variables, limits and continuity, differentiability, gradients, surfaces, maxima and minima, Taylor series, Lagrange multipliers, chain rules, the inverse function theorem and Jacobian derivatives.
2. Integral calculus: double and triple integrals, iterated integrals, Riemann sums, cylindrical and spherical coordinates, change of variables and centre of mass.
3. Vector calculus: line integrals, parametrised surfaces, surface integrals, del, divergence and curl, Stokes' theorem, Green's theorem in the plane, and applications to fluid dynamics and electrodynamics.

That is a complete standard Calculus 3 course, including the vector calculus that some universities teach separately.

MATH2011 topic map

MATH2011 vs MATH2111

The two courses list nearly the same topics and exclude each other. Engineering students often take engineering mathematics courses instead, such as MATH2018 or MATH2019, which cover partial differentiation, double integrals and an introduction to vector field theory alongside other topics. If Mathematics 1B feels distant, review the MATH1231 guide and the MATH1131 guide.

Example problems at the MATH2011 level

1. Classifying a critical point. For f(x, y) = x^2 + xy + y^2, f_x = 2x + y and f_y = x + 2y are both zero only at (0, 0). With f_{xx} = 2, f_{yy} = 2 and f_{xy} = 1, the discriminant is D = 2 \cdot 2 - 1^2 = 3 > 0 and f_{xx} > 0, so (0, 0) is a local minimum.

2. Divergence and curl. For \mathbf{F} = \langle xy, yz, zx \rangle:

3. A spherical integral. The volume of a ball of radius a is

How to do well in MATH2011

• Master the gradient early. It drives directional derivatives, tangent planes and Lagrange multipliers. Our gradient guide ties them together.
• Sketch regions and surfaces before setting up any integral, and practice reordering with the switching the order lesson.
• Learn Green, Stokes and the Divergence Theorem as one idea: a boundary integral equals an interior integral of a derivative. Compare the Green's theorem and Stokes' theorem pages side by side.
• Keep the Jacobian factors for polar, cylindrical and spherical coordinates on your formula sheet, and rehearse with timed practice sets.

Tip: if a line integral looks messy, compute the curl first. A zero curl on a simply connected region means the field is conservative, and the potential function does the rest.