# Unit Information for MATH2961: Linear Algebra and Vector Calculus (Advanced)

## Time and Place

There are four lectures and one tutorial per week. See the timetable for times and venues.

## Lecturers

• Alexander Fish (Vector Calculus)
Consultation: Wednesday 3pm, Carslaw 712 (first half of semester)
• Andrew Mathas (Linear Algebra)
Consultation: Wednesday 11-12, Carslaw 718 (second half of semester)

Questions about MATH2961 shoud be posted on the EdStem discussion board (the EdStem page will be constructed in the first week of the semester) Confidential questions, which include your name and SID, can be sent be emailed to MATH2961@sydney.edu.au. Questions sent to this email address will be redirected to EdStem if they are of general interest. Anonymous emails may not receive a reply.

## Tutorials

You should attend at the time and place given on your timetable. See the timetable for the tutorial times. Please always take your lecture notes to the tutorial, so you can look up what you need to solve the problems.

Tutorial sheets are available online as PDF files from the resources page on the Friday of the week before the tutorial takes place. No hard copies will be distributed.

Tutorials are an integral part of the course. You can only learn mathematics (or any other subject) by doing problems yourself, so attending tutorials is absolutely essential for performing well in the course.

## Assessment

There will be two assignments worth 10% each, two 40 min quizzes worth 10% each. The final exam counts 60% of the total assessment.

Quiz Dates:
Quiz 1: Tuesday 4th April (in lecture)
Quiz 2: Tuesday 16th May (in lecture)
Assignment Collection Dates:
Assignment 1: Monday 24th April
Assignment 2: Monday 29th May

All assignments must be submitted electronically as PDF files using TurnItIn in Blackboard. Late assignments will not be marked.

If it is favourable to the majority of students the Better Mark Principle from first year may be used.

Final Exam: There will be a two-hour final exam. Only material covered in lectures and tutorials will be tested using questions addressing the outcomes. The exam will also contain questions on the theory and proofs, and not just problems to solve.

For all assessments, the rules for special consideration/arrangement apply. The maximal possible extension is 7 days.

The final mark is determined by the following criteria

• High Distinction (HD), 85–100: representing complete or close to complete mastery of the material;
• Distinction (D), 75–84: representing excellence, but substantially less than complete mastery;
• Credit (CR), 65–74: representing a creditable performance that goes beyond routine knowledge and understanding, but less than excellence;
• Pass (P), 50–64: representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.
• Fail (F), 0–49: representing rather limited understanding on a significant range of topics and concepts.

## Course Description

For the official description see the MATH2961 handbook entry in the central unit of study handbook.

The first part of the unit is vector calculus. In this part we further develop the multivariable calculus from first year. Emphasis is on multiple integrals, maxima and minima of functions of several variables and the integral theorems of vector calculus. This includes line and surface integrals, and the theorems of Gauss, Green and Stokes.

The second part of the unit is an introduction to abstract linear algebra, building on the linear algebra from first year. We introduce fields and vector spaces and prove some basic properties of linear transformations. We define the notion of linear independence and bases and look at the classification of linear operators. In the last part we look at inner product spaces, unitary operators and orthogonal diagonalisation.

Assumed Knowledge: A working knowledge of the calculus and linear algebra components of advanced first year mathematics.

## Outcomes:

Students who successfully complete the vector calculus part of this course should

• have a solid understanding of the geometry of Euclidean space, and be comfortable working with the scalar and vector product;
• be comfortable with functions of several real variables, and be able to work with limits involving such functions;
• be able to compute partial derivatives and directional derivatives, and be able to apply differential calculus to investigate the maxima and minima of functions of several real variables;
• be able to compute multiple integrals, line integrals, and surface integrals;
• be able to use multiple, line and surface integrals to compute area, volume, arc length and surface area;
• be able to apply the transformation formula (change of variables) for multiple integrals;
• appreciate the geometry of multiple integrals, line integrals, and surface integrals;
• be able to parametrise arcs and surfaces;
• understand the significance of the integral theorems of Green, Gauss and Stokes, and be able to apply them;

Students who successfully complete the linear algebra part of this course should
• be familiar with the abstract definitions of fields and vector spaces, and the standard examples such as $$\mathbb Q$$, $$\mathbb R$$, $$\mathbb C$$, $$\mathbb Z_p$$ with $$p$$ prime;
• know the definition of and be able to work with linear independent sets in vector spaces;
• know the notion of bases and dimension in a finitely generated vector space;
• know basic properties of linear operators;
• be able to find the matrix representations of linear operators with respect to given bases;
• be able to perform changes of basis, in particular to find diagonal or block matrix representations;
• have a working knowledge of determinants;
• understand inner products, orthogonal matrices and diagonalisation of symmetric matrices;
• be able to apply the knowledge of the above concepts to a variety of concrete and more abstract examples.

## Lecture Notes

Vector Calculus by D Daners and Linear Algebra by D Daners (two separate books) available from Kopystop. Also make use of the follwing reference books (you do not need to buy any of these).

## Reference Books

• S. Axler, Linear Algebra done Right (Springer, NY 1997 Library 512.5/372
• C. W. Curtis, Linear Algebra: an introductory approach, 4th ed. (Springer-Verlag, N.Y. 1984). Library 512.897/148
• P. Halmos, Finite-dimensional Vector Spaces, 2nd ed. (Van Nostrand, Princeton, 1958). Library copies
• M. O'Nan, Linear Algebra, 2nd ed. (Harcourt, Brace, Jovanovich, N.Y. 1976). Library 512.897/55
• T. M. Apostol, Mathematical Analysis: a modern approach to advanced calculus (Addison-Wesley Publishing Company Inc., Reading, Mass., 1957).
• T. M. Apostol, Calculus, Vol II: multivariable calculus and linear algebra with applications to differential equations and probability, 2nd ed. (Blaisdell Publishing Co., Ginn and Co., Waltham, Mass. – Toronto, Ont. – London, 1969). (Library 517 43 A)
• E. Kreyszig, Introduction to differential geometry and Riemannian geometry (University of Toronto Press, Toronto, Ont., 1968).
• Schaum's outline of theory and problems of advanced calculus (Library 517 132)