MATH203 - Calculus of Several Variables

Year

Credit points

Campus offering

Prerequisites

MATH104 Differential and Integral Calculus and MATH107 Introduction to Logic and Algebra

Teaching organisation

Unit rationale, description and aim

Building on the introductory calculus unit MATH104, this unit introduces students to the calculus of functions of several variables. This will enable students to consider, and solve problems dealing with, situations where continuous changes in several factors influence the final outcome. 

This unit extends the concepts of differentiation and integration from functions of a single variable to functions of more than one variable, introducing partial derivatives and iterated integration including line and surface integrals. Simple ordinary differential equations will also be considered. The aim of this unit is to extend students’ knowledge to include multiple dimensions and the calculus of functions involving many dimensions and to introduce some basic analytic ideas.

Learning outcomes

To successfully complete this unit you will be able to demonstrate you have achieved the learning outcomes (LO) detailed in the below table.

Each outcome is informed by a number of graduate capabilities (GC) to ensure your work in this, and every unit, is part of a larger goal of graduating from ACU with the attributes of insight, empathy, imagination and impact.

Explore the graduate capabilities.

On successful completion of this unit, students should be able to:

LO1 - Find all the partial derivatives of a function of several variables (GA4, GA5, GA8, GA10) 

LO2 - Find the extreme points of a function of several variables given the domain of the function (GA4, GA5, GA8) 

LO3 - Find the extreme points of a function subject to a given constraint (GA4, GA5, GA8)

LO4 - Use iterated integration techniques to evaluate definite integrals of functions of several variables, using Fubini’s theorem and/or polar coordinates if necessary (GA4, GA5, GA8, GA10) 

LO5 - Calculate areas and volumes using iterated integrals (GA4, GA5, GA8, GA10) 

LO6 - Evaluate Line and Surface integrals using Green’s Theorem and related results (GA4, GA5, GA8)

Graduate attributes

GA4 - think critically and reflectively 

GA5 - demonstrate values, knowledge, skills and attitudes appropriate to the discipline and/or profession 

GA8 - locate, organise, analyse, synthesise and evaluate information 

GA10 - utilise information and communication and other relevant technologies effectively.

Content

Topics may include: 

1. Functions of several variables, level curves and cross-sections. 
2. Partial derivatives, tangent planes, differentials, estimating functions using differentials. 
3. Second order partial derivatives and the mixed derivative theorem, total derivative rule and other chain rules, implicit differentiation, stationary points and their nature. 
4. Review of vectors; vectors and calculus including vector fields. 
5. The method of Lagrange multipliers to find extreme points 
6. Integration of functions of several variables. 
7. Iterated integrals, volumes, changing the order of integration, Fubini’s theorem. 
8. Changing integrals from Cartesian to polar coordinates. 
9. Line and Surface Integrals, div, grad and curl. 
10. Green’s Theorem, Stokes’ Theorem and Divergence Theorem 

Learning and teaching strategy and rationale

As is common in Mathematics a variety of Active Learning approaches promote the best acquisition of skills and understanding. Lectures will typically be structured to provide explanations of the material to be covered, along with examples of the applications of that material, as well as formal and informal tasks for students that will reinforce that learning. Splitting the taught content of the lectures into short topics will ensure increased student engagement with the material, as well as providing opportunities for students to consolidate their learning before new material is covered. This allows students to learn the skills and then build understanding, competence and confidence via (ideally, face-to-face) tutorials involving cooperative groups, peer review and other relevant strategies. In all cases this should be supported using available online technology. 

This unit will normally include the equivalent of 24 hours of lectures (typically 2 hours per week for 12 weeks) together with 24 hours attendance mode tutorials. Lectures will also be recorded and, where possible or required, students may have access to an online tutorial. 150 hours in total with a normal expectation of 48 hours of directed study and the total contact hours should not exceed 48 hours. The balance of the hours becoming private study 

Assessment strategy and rationale

To successfully complete an undergraduate Mathematics sequence, students need an understanding of a variety of basic Mathematical topics and an ability to apply that understanding to a variety of problems. To succeed at problem solving in Mathematics, students must have these skills at their fingertips and be able to recall them and choose an appropriate approach under some pressure. The assessment strategy chosen, while traditional, tests and supports student learning by providing opportunities to develop and test their problem-solving skills through the unit.

The continuous assessment component allows for the early detection of problems a student might be having and so ensures that appropriate guidance can be given early enough to support later learning in the unit. Students will be required to submit responses to questions dealing with simple problems in calculus of several variables. As specified times, through the semester, students will submit responses to some of the questions to allow for the provision of feedback and learning support to students. The examination components ensure that students have fully integrated the learning and can bring a variety of strategies to bear under pressure. 

Typing of Mathematical notation either requires a significant investment of time, or knowledge of advanced Mathematical typesetting software. Neither of those skills are suitable for an undergraduate course in Mathematics. Consequently, assignments, tests and examinations are typically handwritten and either submitted as hardcopy or scanned and submitted to a dropbox, rather than through Turnitin. Students may choose to type their responses and submit them electronically, but this is not a requirement of the unit. 

Overview of assessments

Representative texts and references

Anton H, Bivens I, Davis S (2016) Calculus: Early Transcendentals, 11th Edition New York: John Wiley & Sons. 

Betounes, D & Betounes, M.R. (2019) Calculus: Concepts and Computation, 3rd Edition Kendall/Hurt Publications. 

Courant, R., Robbins, H. & Stewart, I. (1996) What is Mathematics? Oxford: Oxford University Press. 

Edwards, C.H. & Penney D.E. (2007) Calculus and early trancendentals. Prentice-Hall Inc. 

Hughes-Hallett, D. et al (2005) Calculus Single and Multivariable, New York: John Wiley & Sons 

Spivak, M. (2008) Calculus, 4th Edition, Publish or Perish 

Stein, S.K. & Barcellos, A. (1992) A Calculus and Analytic Geometry, New York: McGraw Hill 

Strang, G. (2017) Calculus 3rd Edition, Wellesley-Cambridge