35102 Introduction to Analysis and Multivariable Calculus

Warning: The information on this page is indicative. The subject outline for a particular semester, location and mode of offering is the authoritative source of all information about the subject for that offering. Required texts, recommended texts and references in particular are likely to change. Students will be provided with a subject outline once they enrol in the subject.

Handbook description

Many problems in business and science involve the study of systems with many interrelated (examples include the study of investment portfolios and the analysis of ecological systems). This subject introduces students to the principles of calculus for functions of several variables required in such applications, and to the theoretical foundations of mathematics that underpin them. Topics include vectors; products of vectors; equations of lines and planes; functions of several variables; partial derivatives and gradient; double integrals; sequences and their convergence; continuous and uniformly continuous functions; properties of continuous functions on a closed interval; differentiability; power series, tests for convergence and radius of convergence; Taylor and Maclaurin series; sequences and series of functions and Weierstrass M-test; upper and lower sums; and the Riemann integral.

Subject objectives/outcomes

Students who successfully complete this subject should be able to:

1. perform routine calculations in multivariable calculus and elementary analysis
2. understand the methods of proof of the theorems and be able to prove key results
3. use formal and informal language to demonstrate understanding of the underlying concepts and theorems
4. communicate the above knowledge clearly, logically and critically
5. use the computer algebra system Mathematica to perform calculations and explore mathematical ideas relevant to the subject content
6. be able to apply the subject matter covered in lectures, tutorials and assignments to previously unseen problems and proofs
7. respond ethically and appropriately to the completion of learning activities and assessment tasks.
   

Contribution to course aims and graduate attributes

The Faculty of Science has determined that its courses will aim to develop the following attributes in students at the completion of their course of study. Each subject will contribute to the development of these attributes in ways appropriate to the subject and the stage of progression, thus not all attributes are expected to be addressed in all subjects.

1. Disciplinary knowledge and its appropriate application
2. An inquiry-oriented approach
3. Professional skills and their appropriate application
4. The ability to be a lifelong learner
5. Engagement with the needs of society
6. Communication skills
7. Initiative and innovative ability

This subject introduces the formal language of mathematical proof required in more advanced mathematical discourse and develops practical skills in multivariable calculus. In this way, students are equipped for later subjects in the degree program, such as 35231 Differential Equations and 35252 Advanced Calculus and the subjects that follow on from these; students also acquire foundational skills needed by the professional mathematician for ongoing learning. Homework and tutorial questions are designed to help students develop skills in problem solving and critical thinking whilst the regular weekly assessment tasks are designed to encourage the development of personal organisational skills and time management skills in addition to the practical skills of this subject. Computer laboratory activities are designed to aid visualisation and concept development.

Teaching and learning strategies

Spring Session:

Weekly on campus:

• Two 1.5 hr lectures
• 1 hr tutorial
• 1 hr laboratory

Face-to-face classes will incorporate a range of teaching and learning strategies including short presentations, discussion of readings and student groupwork. The five hours of classes each week are supported by at least five hours per week of individual or group study, developing and practising skills by doing many textbook questions etc. Students may use the online resources available at external links on UTSOnline to obtain further insight.

Summer Session:

Face-to-face classes will incorporate a range of teaching and learning strategies including short presentations, discussion of readings and student groupwork. The nine hours of classes each week are supported by at least nine hours per week of individual or group study, developing and practising skills by doing many textbook questions etc. Students may use the online resources available at external links on UTSOnline to obtain further insight.

Content

Vectors; products of vectors; equations of lines and planes; functions of several variables; partial derivatives and gradient; double integrals; sequences and their convergence; continuous functions; properties of continuous functions on a closed interval; differentiability; power series, tests for convergence and radius of convergence; Taylor and Maclaurin series; sequences and series of functions and Weierstrass M-test; upper and lower sums; the Riemann integral.

Minimum requirements

In order to pass this subject, a student must achieve a final result of 50% or more and achieve 40% or more on the final examination. The final result is simply the sum of all the marks gained in each piece of assessment. Students who obtain 50 marks or more but fail to score 40% or more on the final examination will be given an X grade (fail).

Recommended texts

• Hass, J., Weir, M. D. & Thomas, G. B. University Calculus (Alternate Edition), Pearson Education, 2008. (ISBN 0321500970 or 9314994242614 with MyMathLab code)
• McLelland, G. J. An Introduction to Analysis, UTS, 2011, CN 4565

[Available for $10 at UTS Union Shop, Level 3, Tower Building, Broadway; when sold out, orders are taken on a prepaid basis only. A copy is also kept in UTS Library Closed Reserve.]

Other resources

• Brannan, D. A First Course in Mathematical Analysis, Cambridge University Press, 2006.
• Houston, K. How to Think Like a Mathematician, Cambridge University Press, 2009.
• Kosmala, W. A. J. A Friendly Introduction to Analysis: Single and Multivariable, 2nd edition, Pearson Prentice Hall, 2004.