MTH2015 · Multivariable calculus (advanced)

Overview

This unit is an alternative to MTH2010 for students with a strong mathematical foundation. Students enrolled in MTH2015 will follow the same curriculum as students in MTH2010 and will cover additional more advanced material. Functions of several variables, partial derivatives, extreme values, Lagrange multipliers. Multiple integrals, line integrals, surface integrals. Vector differential calculus; grad, div and curl. Integral theorems of Gauss and Stokes. Use of a computer algebra package. Curves in 3-space, notions of torsion and curvature. Introductory notions of topology and geometry (stereographic projection). Basic introduction to real analysis: pointwise versus uniform convergence of functions of one variable. Introduction to complex analysis: holomorphic functions, harmonic functions, complex integration, Cauchy's integral formula, the fundamental theorem of Algebra.

Areas of study: Applied mathematics Astrophysics Climate and atmospheric science Financial and insurance mathematics Mathematical statistics Mathematics

Offerings

Assessment

The Handbook lists an examination for this unit.

Assessment details may change. Please refer to the assessment information in Moodle closer to the start of the teaching period.

Requisites

The Handbook lists no prerequisite, corequisite or prohibition for this unit.

Learning outcomes

1. Apply multivariable calculus techniques to solve problems in mathematics, physics, and related disciplines;
2. Explain the definition of the derivative of a multivariable function and its geometric interpretation as the tangent plane, as well as its relation to the gradient and directional derivative;
3. Find and classify extrema of functions of several variables, including constrained optimisation using Lagrange multipliers;
4. Evaluate and interpret line, surface, and volume integrals in Cartesian, cylindrical, and spherical coordinates, and apply the integral theorems of Green, Gauss, and Stokes;
5. Communicate mathematical reasoning effectively, both orally and in writing, and collaborate to analyse and solve problems using multivariable calculus techniques;
6. Extend and deepen understanding of multivariable calculus through advanced theoretical concepts, rigorous proofs, and exposure to additional topics such as curvature and torsion, stereographic projection, uniform convergence, and analytic functions.

Workload

• Three 1-hour seminars; • One 1-hour seminar; • One 2-hour applied class (in weeks 2-12) and • 6 hours of independent study per week.

Student experience, not official rules. Nothing here changes what the Handbook says.