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.

Requisites

Learning outcomes

1. Apply multivariable calculus to problems in the mathematical and physical sciences;
2. Find and classify the extrema of functions of several variables;
3. Compute Taylor series for functions of several variables;
4. Compute line, surface and volume integrals in Cartesian, cylindrical and polar coordinates;
5. Apply the integral theorems of Green, Gauss and Stokes;
6. Use computer algebra packages to solve mathematical problems;
7. Present a mathematical argument in written form;
8. Understand and apply the formal definition of a limit to functions of several variables;
9. Prove various identities between grad, div and curl;
10. Develop and present rigorous mathematical proofs.
11. Demonstrate an understanding of the notions of torsion and curvature and be able to compute them;
12. Apply stereographic projection and its properties;
13. Articulate the difference between pointwise and uniform convergence for functions of one variable;
14. Use the properties of analytic functions to prove fundamental results.

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.