# MTH2015 - Multivariable calculus (advanced) - 2018

## 6 points, SCA Band 2, 0.125 EFTSL

Refer to the specific census and withdrawal dates for the semester(s) in which this unit is offered.

Faculty

Science

Organisational Unit

School of Mathematical Sciences

Chief examiner(s)

Coordinator(s)

Unit guides

Offered

Clayton

• Second semester 2018 (On-campus)

Prerequisites

A High Distinction in VCE Enhancement Mathematics or MTH1030; a Distinction in MTH1035; or by approval of the Head of School of Mathematical Sciences. In order to enrol in this unit students will need to apply via the Science Student Services officeScience Student Services office (https://www.monash.edu/science/current-students/admissionsstudy-optionsinternal-transfer_arch).

Prohibitions

ENG2005, ENG2091, MTH2010

## Synopsis

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.

## Outcomes

On completion of this unit students will be able to:

1. Understand and 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.

## Assessment

Examination (3 hours): 60% (Hurdle)

Continuous assessment: 40%

Hurdle requirement: To pass this unit a student must achieve at least 50% overall and at least 40% for the end-of-semester exam.