Multivariable and Complex Calculus II

Resources for Multivariable and Complex Calculus II (MCC)聽- for more information about the courses, please see聽course outlines.

MLC Drop-In Centre

Students from MCC II can use the drop-in centre, but we will give priority to first year students. If you see every other table has students, then we would appreciate it if you found another place to sit so that there is room for new students.

Please note that not every staff member in the MLC knows all of the content in MCC II, there will be times when we can only give general study advice for this course.

Assumed knowledge

Resources that you can use to help you revise assumed knowledge for MCC. The various topics in the course are taught assuming you know this information.

  • Techniques of integration

    A very large part of MCC concerns integration. You need to be really comfortable with performing definite integrals. These resources from Maths IA should help you revise.

    These three PDF handouts are useful for doing different types of integration:

    This PDF document contains many written worked examples of various techniques of integration.

    This seminar was given in Sem 1 2021 and covered the techniques of integration (substitution, by parts, trig substitution, partial fractions and upper and lower sums).

    This seminar was given in Sem 1 2014 and covered the entire of the techniques of integration topic. (Note that the seminar also includes reduction formulas and numerical integration,聽which are no longer in Maths 1A.)

  • Complex numbers

    The end of MCC concerns the calculus of complex numbers, which requires you to know about complex numbers first!

    In Semester 1 2017, the MLC gave a revision seminar on Complex Numbers to Maths 1M students. In this seminar, David covered where the complex numbers fit in the families of number, and how the various operations relate to the three main representations of complex numbers.聽

  • Vector algebra

    In MCC, it's important to be able to represent objects like curves, lines and planes with vectors and parameters. It's also important to know how to do vector calculations like the dot and cross product.聽These resources made for Maths 1M will help to revise and prepare.

    This seminar in聽Semester 1 2015 was聽on vectors. The seminar covered the major things you can calculate with vectors (such as dot and cross product, projections and angles), and doing geometry proofs with vectors, and also briefly discussed equations of lines and planes. Note I mentioned a list of versions of the right hand rule, which is linked below.

    This seminar in Semester 1 2016 was on curves. David discussed what a curve is, converting between parametric and Cartesian form, and finding tangents to curves.

    This seminar in Semester 2 2020 started with a section on dot and cross products of vectors in 3D.

  • Multivariable calculus

    MCC II picks up roughly where Maths 1B calculus left off, so it might help to revise the multivariable calculus content in Maths IB.

    In this seminar from Summer Semester 2019,聽David spent a lot of time talking about different ways to imagine multivariable functions and their derivatives.

    This seminar from 2012 covers domains, ranges, partial derivatives, maxima and minima in multivariable calculus..

    This seminar from 2013 covered directional derivatives and tangent planes in multivariable (3D) calculus.

Revision seminars relevant to MCC

David has given revision seminars for MCC and other courses which may help to think about the concepts and understand problem-solving techniques. They are organised here in order of time.

2024

Semester 1 2024, David gave a revision seminar discussing change of variables in multivariable integration, and then talked about the four fundamental theorems of multivariable calculus. He mentions a handout from the 2019 seminar, which is also included here.

2023

In Semester 1 2023, David gave a revision seminar discussing聽optimisation on a closed bounded set, and also some ideas about curves. Unfortunately the video was corrupted partway through processing so only the first 20 minutes or so is available. However the notes from the whole seminar are still provided here.

2022

In Semester 1 2022, David gave a revision seminar discussing various requests from students. There were sections on聽suffix notation, choosing limits of integrals, parameterisations, and a little on the residue theorem.

2021

In Semester 1 2021. David gave a revision seminar where he discussed setting up and doing integrals in order to calculate volumes.

  • No聽 notes available from 2021 seminar

2019

In Semester 1 2019, David did聽a revision seminar on integrals. He talked about each of the nine kinds of integrals in the course, comparing their features and finding where the various integral theorems connect them together. He finished off with a couple of examples of using the theorems to make some integrals easier. (David thinks this is one of his best revision seminars for any course ever.) He also turned his paper-and-play-dough diagram from the seminar into a handout, which is linked below.

2018

    In Semester 1 2018,聽David did a revision seminar where he聽talked about path, surface and volume integrals, working through an exam question involving all of them. He then finished off with a description of parameterisations of various curves and surfaces (starting at 1h40m30s).

    2017

    In Semester 1 2017, David did a revision seminar which聽contained examples of doing the path, surface and volume integrals that are mentioned in Stokes' Theorem and Gauss' Divergence Theorem.

    In 2017, David did a fully worked example for an engineering course about calculating the integrals involved to verify Green's Theorem in the plane. (Note this version of Green's Theorem might be slightly different to the one presented in MCC, but the integral techniques are the same.)聽

      2014

      In semester 2, 2014, the MLC gave a revision seminar on Complex Calculus to Engineering Maths IIB聽students. The content is similar to the content in MCC, so it should be helpful for your study. (Please forgive the 30 minutes or so of horrible flailing at the end that went on when I tried to interpret the Eng Maths IIB聽lecturer's example of using Jordan's Lemma).聽

      2013

      In Semester 2, 2013,聽David gave a revision seminar on Vector Calculus including different types of derivatives and integrals (and some aspects of Complex Calculus) to Engineering Maths IIB students. The content is similar to the content in MCC and so should be helpful for studying MCC.聽

      • Notes from 2013 seminars not available

      In Semester 2, 2013,聽David gave a revision seminar with a section on聽Complex Calculus聽to Engineering Maths IIB students (starting at 1h26m). The content is similar to the content in MCC and so should be helpful for studying MCC (though the terminology might be a little different).

      • Notes from 2013 seminar not available