Unit rationale, description and aim
Multivariable calculus plays a foundational role in modelling and analysing complex systems where several variables change simultaneously. From predicting weather patterns to optimising engineering designs and economic systems, the tools of multivariable calculus are essential in science, technology, and mathematics. The unit builds upon students’ understanding from MATH104, extending their knowledge of single variable calculus to the calculus of functions of several variables. Problems involving multivariable functions arise naturally in geometry, physics, biology, economics, and other applied disciplines. This unit equips students to analyse and solve problems involving 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. Topics include partial derivatives, directional derivatives, iterated integrals, and applications such as line and surface integrals. Simple ordinary differential equations will also be considered. The aim of this unit is to provide students with tools to analyse and solve problems involving multivariable systems, deepening their understanding of calculus beyond one dimension. It also introduces foundational analytical concepts that prepare students for advanced mathematical study.
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.
Analyse the behaviour of functions of several vari...
Learning Outcome 01
Determine the extreme points of a multivariable fu...
Learning Outcome 02
Evaluate definite integrals of functions of severa...
Learning Outcome 03
Content
Topics may include:
- Functions of several variables, level curves and cross-sections.
- Partial derivatives, tangent planes, differentials, estimating functions using differentials.
- Second order partial derivatives and the mixed derivative theorem, total derivative rule and other chain rules, implicit differentiation, stationary points and their nature.
- Review of vectors; vectors and calculus including vector fields.
- The method of Lagrange multipliers to find extreme points
- Integration of functions of several variables.
- Iterated integrals, volumes, changing the order of integration, Fubini’s theorem.
- Changing integrals from Cartesian to polar coordinates.
- Line and Surface Integrals.
- Green’s Theorem and related results.
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.
At specified times, through the semester, students will sit two supervised components to ensure that students have fully integrated the learning and can independently bring a variety of strategies to solve a variety of problems. The take home assignment will allow students to develop other skills including traditional and technological problem-solving methods.
Overview of assessments
Assessment Task 1: Take Home Assignment(s) Contin...
Assessment Task 1: Take Home Assignment(s)
Continuous assessment – A single task which is submitted in 1, 2 or 3 parts across the semester as stipulated by the lecturer.
30%
Assessment Task 2: Test Content and application t...
Assessment Task 2: Test
Content and application test 1. This test will be conducted under supervision and on campus.
35%
Assessment Task 3: Test Content and application ...
Assessment Task 3: Test
Content and application test 2. This test will be conducted under supervision and on campus.
35%
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 through tutorials that ideally occur face to face. Face-to-face learning supports immediate feedback, peer interaction, and collaborative problem-solving (essential elements in developing mathematical thinking) while also enabling tutors to identify and address misconceptions early.
Continuous formative learning opportunities allow for the early detection of problems a student might be having and ensures that appropriate guidance can be given early enough to support later learning in the unit. Students will be expected to submit written responses to questions involving core mathematical concepts, methods, and problem-solving strategies. In all cases this should be supported using available online technology.