MATHS 340 : Real and Complex Calculus


2024 Semester Two (1245) (15 POINTS)

Course Prescription

Calculus plays a fundamental role in mathematics, answering deep theoretical problems and allowing us to solve very practical problems. This course 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 major topics in the course are multivariate and complex calculus. The course is required for the applied mathematics pathway in the mathematics major, though other mathematics and physics students will benefit from studying these important topics. 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. Having passed other Stage II mathematics courses or related physics courses, in addition to MATHS 250, will enhance your preparation for 340. 

Course Requirements

Prerequisite: MATHS 250

Capabilities Developed in this Course

Capability 3: Knowledge and Practice
Capability 4: Critical Thinking
Capability 5: Solution Seeking
Capability 6: Communication
Graduate Profile: Bachelor of Science

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 3, 4, 5 and 6)
  2. Demonstrate the ability to calculate double, triple, surface, volume and line integrals. (Capability 3, 4, 5 and 6)
  3. Demonstrate how to use Green's and Stokes' theorems and the divergence theorem on simple examples. (Capability 3, 4, 5 and 6)
  4. Demonstrate a good knowledge of complex arithmetic and using functions of a complex variable. (Capability 3, 4, 5 and 6)
  5. Demonstrate how to use the Residue Theorem to evaluate specific types of contour and improper integrals. (Capability 3, 4, 5 and 6)


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

A minimum of 35% on the final exam is required to pass the course.


Tuākana Science is a multi-faceted programme for Māori and Pacific students providing topic specific tutorials, one-on-one sessions, test and exam preparation and more. Explore your options at

Whanaungatanga and manaakitanga are fundamental principles of our Tuākana Mathematics programme which provides support for Māori and Pasifika students who are taking mathematics courses. The Tuākana Maths programme consists of workshops and drop-in times, and provides a space where Māori and Pasifika students are able to work alongside our Tuākana tutors and other Māori and Pasifika students who are studying mathematics.

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.

Workload Expectations

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 each week of 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.

Delivery Mode

Campus Experience

  • Attendance is expected at scheduled activities, including tutorials, to complete components of the course.
  • Lectures will be available as recordings. Other learning activities, including tutorials, will not be available as recordings.

Learning 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.

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.

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 feedback is considered on an on-going basis throughout the year.

Academic Integrity

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 for potential plagiarism or other forms of academic misconduct, using computerised detection mechanisms.

Class Representatives

Class representatives are students tasked with representing student issues to departments, faculties, and the wider university. If you have a complaint about this course, please contact your class rep who will know how to raise it in the right channels. See your departmental noticeboard for contact details for your class reps.


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.

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

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

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

Learning Continuity

In the event of an unexpected disruption, we undertake to maintain the continuity and standard of teaching and learning in all your courses throughout the year. If there are unexpected disruptions the University has contingency plans to ensure that access to your course continues and course assessment continues to meet the principles of the University’s assessment policy. Some adjustments may need to be made in emergencies. You will be kept fully informed by your course co-ordinator/director, and if disruption occurs you should refer to the university website for information about how to proceed.

The delivery mode may change depending on COVID restrictions. Any changes will be communicated through Canvas.

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


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 students may be asked to submit 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. In exceptional circumstances changes to elements of this course may be necessary at short notice. Students enrolled in this course will be informed of any such changes and the reasons for them, as soon as possible, through Canvas.

Published on 31/10/2023 10:53 a.m.