# Calculus and Applications 2 KMA252

### Introduction

This unit is a continuation of KMA152 and KMA154, with emphasis on the application of multivariable calculus and Fourier Series to problems in mathematics, the physical and biological sciences, economics, and engineering. The calculus section of this unit is focussed on dealing with functions of several variables, of which the typical case is z = f(x,y). Functions like this are important because they describe many of the situations we encounter when applying mathematics to models of the real world. The graph of the function z = f(x,y) is a surface, so it might be used to describe building structures; aeroplane wings; temperature, stress, and pressure distributions; income as a function of various expenses; and so on. We need to be able to say how rapidly such a surface curves, and that immediately requires us to do calculus on functions of two (or more) variables. We will also need to consider vectors that are functions of several variables. Some obvious examples of these are the velocity vector in a moving fluid, the heat-flow vector in a solid, and the electric and magnetic fields produced by an antenna. This will lead us to consider more advanced concepts such as circulation, compressibility, divergence, and curl. Understanding this material is fundamental to the study of all areas of Engineering and (continuum) Applied Mathematics, and it underpins modern continuum mechanics and electromagnetic theory. Topics will be introduced in the Cartesian (rectangular) coordinate system but we will also investigate functions, regions, and vectors defined in cylindrical and spherical coordinates. The Fourier-Series section of this unit is concerned with how to represent periodic functions. Previously we have looked at power series as an infinite sum of terms involving increasing powers of a particular variable. A Fourier series is an infinite sum of terms involving sine and cosine functions. This is an important concept for solving problems in acoustics, signal processing, heat-flow theory, fluid mechanics, vibrations, electromagnetic field theory, and so on.

### Summary

Unit name Calculus and Applications 2
Unit code KMA252
Credit points 12.5
College/School College of Sciences and Engineering
School of Natural Sciences
Discipline Mathematics
Coordinator Doctor Michael Brideson
Available as an elective? Yes
Delivered By University of Tasmania
Level Intermediate

### Availability

Location Study period Attendance options Available to
Hobart Semester 1 On-Campus International Domestic
Online Semester 1 Off-Campus International Domestic

#### Key

On-campus
Off-Campus
International students
Domestic students
##### Note

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#### Key Dates

Study Period Start date Census date WW date End date
Semester 1 26/2/2024 22/3/2024 15/4/2024 2/6/2024

* The Final WW Date is the final date from which you can withdraw from the unit without academic penalty, however you will still incur a financial liability (refer to How do I withdraw from a unit? for more information).

Unit census dates currently displaying for 2024 are indicative and subject to change. Finalised census dates for 2024 will be available from the 1st October 2023. Note census date cutoff is 11.59pm AEST (AEDT during October to March).

### Learning Outcomes

• interrogate the behaviour of multivariable functions using a variety of analytical techniques
• manipulate functions, operations, and problems into different coordinate systems and solve problems accordingly
• solve unconstrained and constrained optimisation problems involving several variables
• construct and solve double and triple integrals, as well as line, surface, and volume integrals
• develop and use vector calculus techniques to solve problems involving scalar fields and vector fields
• develop and use Fourier Series techniques for periodic functions

### Fee Information

The 2024 Commonwealth Supported Place (CSP) rates are still being finalised by the Government and we will update the domestic fee information as soon as we have more details.

### Requisites

#### Prerequisites

(KMA152 OR JEE103) AND (KMA154 OR JEE104)

#### Mutual Exclusions

You cannot enrol in this unit as well as the following:

KME771 AND KME271

### Teaching

Teaching Pattern 3x1-hour workshops and 1-hour tutorial weekly. Online Homeworks (20%)|Final Examination (40%)|Written Assignments (40%) View the lecture timetable | View the full unit timetable

#### Textbooks

Required Required readings will be listed in the unit outline prior to the start of classes. Thomas’ Calculus, 14th edition in SI Units (electronic), Hass et al. (Pearson Education, 2020). Students who completed KMA152 and KMA154 should have access to an online version of the text above. Advanced Engineering Mathematics, 9th edition, E. Kreyszig (Wiley 2006) Advanced Engineering Mathematics, 3rd edition, D Zill and M Cullen (Jones and Bartlett 2006) Advanced Engineering Mathematics, 8th edition, P O’Neil (Cengage 2018) The Advanced Engineering Mathematics texts are excellent reference books, covering a plethora of applied mathematics topics including linear algebra, vector calculus, differential equations (ordinary and partial), Fourier analysis, complex analysis, numerical analysis, optimisation, and probability and statistics. There are many topics that you will meet in other units in Mathematics, Physics, and Engineering. There is quite a large number of text-books with the same or similar title, and you should be able to find some of these in the library. Booktopia textbook finder

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