# MATHS 340 : Real and Complex Calculus

## Science

### Course Prescription

Calculus plays a fundamental role in mathematics, answering deep theoretical problems and allowing us to solve very practical problems. Extends the ideas of calculus to two and higher dimensions, showing how to calculate integrals and derivatives in higher dimensions and exploring special relationships between integrals of different dimensions. It also extends calculus to complex variables. Recommended preparation: MATHS 253.

### Course Overview

The course is required for the applied mathematics pathway in the mathematics major. The major topics in the course are multivariate and complex calculus. These are important topics and other mathematics students and physics students will benefit from taking the course. For students continuing on to a postgraduate degree in applied mathematics, most postgraduate courses in applied mathematics have a prerequisite of at least a B- in 340. The course has a prerequisite of MATHS 250. This is the minimum preparation for the course. Having passed other Stage II mathematics courses or related physics courses will enhance your preparation for 340.

### Course Requirements

Prerequisite: MATHS 250

### Capabilities Developed in this Course

 Capability 1: Disciplinary Knowledge and Practice Capability 2: Critical Thinking Capability 3: Solution Seeking Capability 4: Communication and Engagement

### Learning Outcomes

By the end of this course, students will be able to:
1. Demonstrate the ability to calculate time derivatives in non-Cartesian coordinates. (Capability 1, 2, 3 and 4)
2. Demonstrate the ability to calculate double, triple, surface, volume and line integrals (Capability 1, 2, 3 and 4)
3. Demonstrate how to use Green's and Stokes' theorems and the divergence theorem on simple examples. (Capability 1, 2, 3 and 4)
4. Demonstrate a good knowledge of complex arithmetic and using functions of a complex variable. (Capability 1, 2 and 4)
5. Demonstrate how to use the Residue Theorem to evaluate specific types of contour and improper integrals. (Capability 1, 3 and 4)

### Assessments

Assessment Type Percentage Classification
Assignments 30% Individual Coursework
Test 20% Individual Test
Final Exam 50% Individual Examination
1 2 3 4 5
Assignments
Test
Final Exam

### Tuākana

For more information and to find contact details for the MATHS 340 Tuākana coordinator, please see https://www.auckland.ac.nz/en/science/study-with-us/maori-and-pacific-at-the-faculty/tuakana-programme.html

### Key Topics

1. Time derivatives in plane polar, cylindrical and spherical coordinates.
2. Double and triple integrals.
3. Surface and volume integrals.
4. Divergence theorem in simple applications.
5. Stokes' and Green's theorem in simple applications.
6. Complex arithmetic and functions of a complex variable.
7. Laurent and Taylor series.
8. Evaluating complex and improper integrals using the Residue Theorem.

### Learning Resources

Mathematical Methods for Engineers and Scientists by K.-T. Tang. Available online from Springerlink through the library. There are three volumes. Volumes 1 and 2 will be useful for 340.

Vol 1: Complex Analysis, Determinants and Matrices,
Vol 2: Vector Analysis, Ordinary Differential Equations and Laplace Transforms.

Lecture notes will be made available on Canvas.

### Special Requirements

There are no special requirements.

This course is a standard 15 point course and students are expected to spend 10 hours per week involved in each 15 point course that they are enrolled in.

For this course, you can expect 3 hours of lectures, a 1 hour tutorial, 3 hours of reading and thinking about the content and 3 hours of work on assignments and/or test preparation.

### Digital Resources

Course materials are made available in a learning and collaboration tool called Canvas which also includes reading lists and lecture recordings (where available).

Please remember that the recording of any class on a personal device requires the permission of the instructor.

The content and delivery of content in this course are protected by copyright. Material belonging to others may have been used in this course and copied by and solely for the educational purposes of the University under license.

You may copy the course content for the purposes of private study or research, but you may not upload onto any third party site, make a further copy or sell, alter or further reproduce or distribute any part of the course content to another person.

The University of Auckland will not tolerate cheating, or assisting others to cheat, and views cheating in coursework as a serious academic offence. The work that a student submits for grading must be the student's own work, reflecting their learning. Where work from other sources is used, it must be properly acknowledged and referenced. This requirement also applies to sources on the internet. A student's assessed work may be reviewed against online source material using computerised detection mechanisms.

### Inclusive Learning

All students are asked to discuss any impairment related requirements privately, face to face and/or in written form with the course coordinator, lecturer or tutor.

Student Disability Services also provides support for students with a wide range of impairments, both visible and invisible, to succeed and excel at the University. For more information and contact details, please visit the Student Disability Services’ website at http://disability.auckland.ac.nz

### Special Circumstances

If your ability to complete assessed coursework is affected by illness or other personal circumstances outside of your control, contact a member of teaching staff as soon as possible before the assessment is due.

If your personal circumstances significantly affect your performance, or preparation, for an exam or eligible written test, refer to the University’s aegrotat or compassionate consideration page: https://www.auckland.ac.nz/en/students/academic-information/exams-and-final-results/during-exams/aegrotat-and-compassionate-consideration.html.

This should be done as soon as possible and no later than seven days after the affected test or exam date.

### Student Feedback

During the course Class Representatives in each class can take feedback to the staff responsible for the course and staff-student consultative committees.

At the end of the course students will be invited to give feedback on the course and teaching through a tool called SET or Qualtrics. The lecturers and course co-ordinators will consider all feedback.

Your feedback helps to improve the course and its delivery for all students.

### Student Charter and Responsibilities

The Student Charter assumes and acknowledges that students are active participants in the learning process and that they have responsibilities to the institution and the international community of scholars. The University expects that students will act at all times in a way that demonstrates respect for the rights of other students and staff so that the learning environment is both safe and productive. For further information visit Student Charter (https://www.auckland.ac.nz/en/students/forms-policies-and-guidelines/student-policies-and-guidelines/student-charter.html).

### Disclaimer

Elements of this outline may be subject to change. The latest information about the course will be available for enrolled students in Canvas.

In this course you may be asked to submit your coursework assessments digitally. The University reserves the right to conduct scheduled tests and examinations for this course online or through the use of computers or other electronic devices. Where tests or examinations are conducted online remote invigilation arrangements may be used. The final decision on the completion mode for a test or examination, and remote invigilation arrangements where applicable, will be advised to students at least 10 days prior to the scheduled date of the assessment, or in the case of an examination when the examination timetable is published.

Published on 06/07/2020 11:26 a.m.