MATHS 7101 - Multivariable and Complex Calculus

North Terrace Campus - Semester 1 - 2020

The mathematics required to describe most "real life" systems involves functions of more than one variable, so the differential and integral calculus developed in a first course in Calculus must be extended to functions of more variables. In this course, the key results of one-variable calculus are extended to higher dimensions: differentiation, integration, and the link between them provided by the Fundamental Theorem of Calculus are all generalised. The machinery developed can be applied to another generalisation of one-variable Calculus, namely to complex calculus, and the course also provides an introduction to this subject. The material covered in this course forms the basis for mathematical analysis and application across an extremely broad range of areas, essential for anyone studying the hard sciences, engineering, or mathematical economics/finance. Topics covered are: introduction to multivariable calculus; differentiation of scalar- and vector-valued functions; higher-order derivatives, extrema and Lagrange multipliers; integration over regions, volumes, paths and surfaces; Green's, Stokes' and Gauss's theorems; an introduction to complex numbers and functions; complex differentiation; complex integration and Cauchy's theorems.

  • General Course Information
    Course Details
    Course Code MATHS 7101
    Course Multivariable and Complex Calculus
    Coordinating Unit School of Mathematical Sciences
    Term Semester 1
    Level Postgraduate Coursework
    Location/s North Terrace Campus
    Units 3
    Contact Up to 3.5 hours per week
    Available for Study Abroad and Exchange Y
    Assumed Knowledge MATHS 1012
    Course Description The mathematics required to describe most "real life" systems involves functions of more than one variable, so the differential and integral calculus developed in a first course in Calculus must be extended to functions of more variables. In this course, the key results of one-variable calculus are extended to higher dimensions: differentiation, integration, and the link between them provided by the Fundamental Theorem of Calculus are all generalised. The machinery developed can be applied to another generalisation of one-variable Calculus, namely to complex calculus, and the course also provides an introduction to this subject. The material covered in this course forms the basis for mathematical analysis and application across an extremely broad range of areas, essential for anyone studying the hard sciences, engineering, or mathematical economics/finance.

    Topics covered are: introduction to multivariable calculus; differentiation of scalar- and vector-valued functions; higher-order derivatives, extrema and Lagrange multipliers; integration over regions, volumes, paths and surfaces; Green's, Stokes' and Gauss's theorems; an introduction to complex numbers and functions; complex differentiation; complex integration and Cauchy's theorems.
    Course Staff

    Course Coordinator: Associate Professor Sanjeeva Balasuriya

    Course Timetable

    The full timetable of all activities for this course can be accessed from Course Planner.

  • Learning Outcomes
    Course Learning Outcomes
    1. Demonstrate understanding of the basic concepts of calculus involving more than one real variable.
    2. Demonstrate understanding of the basic concepts of calculus for one complex variable.
    3. Be able to state and apply the major results in the course.
    4. Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty.
    5. Demonstrate skills in communicating mathematics orally and in writing.
    University Graduate Attributes

    This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

    University Graduate Attribute Course Learning Outcome(s)
    Deep discipline knowledge
    • informed and infused by cutting edge research, scaffolded throughout their program of studies
    • acquired from personal interaction with research active educators, from year 1
    • accredited or validated against national or international standards (for relevant programs)
    all
    Critical thinking and problem solving
    • steeped in research methods and rigor
    • based on empirical evidence and the scientific approach to knowledge development
    • demonstrated through appropriate and relevant assessment
    all
    Teamwork and communication skills
    • developed from, with, and via the SGDE
    • honed through assessment and practice throughout the program of studies
    • encouraged and valued in all aspects of learning
    3
    Career and leadership readiness
    • technology savvy
    • professional and, where relevant, fully accredited
    • forward thinking and well informed
    • tested and validated by work based experiences
    5
    Self-awareness and emotional intelligence
    • a capacity for self-reflection and a willingness to engage in self-appraisal
    • open to objective and constructive feedback from supervisors and peers
    • able to negotiate difficult social situations, defuse conflict and engage positively in purposeful debate
    5
  • Learning Resources
    Required Resources
    None.
    Recommended Resources
    1. Vector Calculus by J. E. Marsden and A. J. Tromba (Barr Smith Library 517 M364v.5)
    2. Basic Complex Analysis by J. E. Marsden and M. J. Hoffman (Barr Smith Library 517.54 M364b)
    Online Learning
    This course uses MyUni exclusively for providing electronic resources, such as lecture notes, assignment papers, sample solutions, discussion boards, etc. It is recommended that students make appropriate use of these resources. Link to MyUni login page: https://myuni.adelaide.edu.au/webapps/login/
  • Learning & Teaching Activities
    Learning & Teaching Modes
    This course relies on lectures as the primary delivery mechanism for the material.  Lecture attendance is expected of all students.  Lecture
    material will expand on the material provided in concise form in the  printed course notes, providing different examples and more detailed
    explanations.  Students are expected to take notes during the lectures, and use these in conjunction with the printed course notes in their
    study of the material.

    Students should note that while video recordings of the lectures will be made available, this is to be considered a secondary resource.  Lectures will be focused towards  students within the classroom at that instance in time, rather than to
    optimise a good video recording. 

    Tutorials supplement the lectures by providing exercises and example problems to enhance the understanding obtained through lectures. A sequence of written assignments provides assessment opportunities for students to gauge their progress and understanding.


    Workload

    The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

    Activity Quantity Workload Hours
    Lectures 35 87.5
    Tutorials 5 20.5
    Assignments 6 48
    Total 156
    Learning Activities Summary
    Course Outline

    Course material is arranged into five sections with the approximate lecture times for each section indicated below.  The first four sections comprise "Multivariable Calculus," extending single-variable calculus ideas from first-year courses.  The final section deals with the calculus of functions defined on the complex numbers.

    1. Functions of many variables: preliminaries (4 lectures)
    2. Differentiation of multivariable functions (7 lectures)
    3. Integration of multivariable functions (8 lectures)
    4. Fundamental theorems of multivariable calculus (7 lectures)
    5. Complex calculus (9 lectures)
    Tutorials

    Tutorials in weeks 3, 5, 7, 9, 11 will be based on the material covered since the previous tutorial. Tutorial exercises will be distributed in the week before each tutorial.  Detailed solutions will be made available on MyUni the week after the Tutorial.

    Small Group Discovery Experience
    In the tutorials.
  • Assessment

    The University's policy on Assessment for Coursework Programs is based on the following four principles:

    1. Assessment must encourage and reinforce learning.
    2. Assessment must enable robust and fair judgements about student performance.
    3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
    4. Assessment must maintain academic standards.

    Assessment Summary
    Assessment task Task type Due Weighting Learning outcomes
    Examination Summative Examination period 70% All
    Test Summative and formative Week 7 15% All
    Assignments Summative and formative Weeks 2, 4, 6, 8, 10, 12 2.5% each All


    Due to the current COVID-19 situation modified arrangements have been made to assessments to facilitate remote learning and teaching. Assessment details provided here reflect recent updates.

    There will be 6 assignments, due by 3.00pm on Friday (Weeks 2, 4, 8, 10 and 12) and Thursday (Week
    6 – this is because Friday is a public holiday that week). These are to be submitted electronically
    via MyUni, having scanned the solutions. Late assignments will receive a 50% penalty if submitted
    between 3.01pm and 5.00pm on the submission day; any submitted thereafter will receive an automatic
    zero.

    Mid-semester test:
    There will be a mid-semester test, which will be conducted remotely via MyUni on Thursday, 30
    April, from 10.10am onwards. Please ensure that you do not have other activities for the time
    10.00am-12.00noon on this day (the test is planned for no more than 1 hour, but time will need to be allocated
    for other aspects such as scanning – more information will be given later). According to a University directive
    related to tests/examinations in the current emergency climate, the test will not be invigilated. A
    sample test will be released to students one week before the test, to demonstrate what style of questions
    to expect. Additionally, a detailed information sheet on the test will be released.

    Final examination:
    The final examination will be held during the scheduled examination period, but will be administered
    remotely via MyUni. More details on this will be made available closer to the time.
    Assessment Related Requirements
    An aggregate score of 50% is required to pass the course.
    Assessment Detail
    Assessment Item Distributed Due Weighting
    Assignment 1 Week 1 Week 2 2.5%
    Assignment 2 Week 3 Week 4 2.5%
    Assignment 3 Week 5 Week 6 2.5%
    Assignment 4 Week 7 Week 8 2.5%
    Assignment 5 Week 9 Week 10 2.5%
    Assignment 6 Week 11 Week 12 2.5%
    Test (in-class) At lecture on April 30 Week 7 15%

    Students enrolled in all versions of this course are required to take the in-class test at the scheduled lecture time on Thursday, April 30
    (10.10--11.00 am, in Ligertwood 333).
    Submission
    Homework assignments must be submitted on time with a signed assessment cover sheet. Late assignments will not be accepted. Assignments will be returned within two weeks. Students may be excused from an assignment for medical or compassionate reasons. Documentation is required and the lecturer must be notified as soon as possible.
    Course Grading

    Grades for your performance in this course will be awarded in accordance with the following scheme:

    M10 (Coursework Mark Scheme)
    Grade Mark Description
    FNS   Fail No Submission
    F 1-49 Fail
    P 50-64 Pass
    C 65-74 Credit
    D 75-84 Distinction
    HD 85-100 High Distinction
    CN   Continuing
    NFE   No Formal Examination
    RP   Result Pending

    Further details of the grades/results can be obtained from Examinations.

    Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

    Final results for this course will be made available through Access Adelaide.

  • Student Feedback

    The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

    SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

  • Student Support
  • Policies & Guidelines
  • Fraud Awareness

    Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's student’s disciplinary procedures.

The University of Adelaide is committed to regular reviews of the courses and programs it offers to students. The University of Adelaide therefore reserves the right to discontinue or vary programs and courses without notice. Please read the important information contained in the disclaimer.