MATH140 Fundamentals of Modern Mathematics 2

Overview

Continuation from MATH 130 with an emphasis on mathematical thinking and formalisation and their importance for applications of mathematics.

The techniques covered in this paper form the basic tools used to produce mathematical frameworks for modelling quantifiable phenomena. For example, to model the movement of an object through space, we begin with an algebraic structure in which to specify where our object is, and then study how that position changes with time using methods developed in calculus. Many other problems arising in areas such as Economics or Chemistry can be examined mathematically using the same basic principles. For example, we may need to minimise a manufacturing cost, or the time for a chemical reaction to take place, or the effects of river pollution; in each case the techniques used for the minimisation are based on a mixture of tools relying on both algebra and calculus.
This paper aims to develop proficiency with algebra and calculus, both for use in other subjects and in preparation for further study of Mathematics. MATH 140 is the natural continuation of MATH 130, and provides a strong mathematical background to support other subjects as well as forming a necessary prerequisite for progression to 200-level Mathematics.

About this paper

Prerequisite

MATH 130 or MATH 160

Restriction

MATH 170

Schedule C

Arts and Music, Science

Eligibility

This paper should appeal to a wide variety of students, including Mathematics and Statistics majors or those studying Computer Science, Physics, Chemistry, Surveying, Biological Sciences, Genetics or other disciplines with a quantitative component requiring competent manipulation of mathematical formulae and interpretation of mathematical representations of systems.

Contact

robert.vangorder@otago.ac.nz

Teaching staff

Dr Robert A Van Gorder

Paper Structure

MATH140 focuses on both theoretical ideas and concrete methods. Topics of the lectures and lecture notes will be organized according to the following order:

Chapter I. Integral Calculus

• We introduce several special functions which will help us in our quest to calculate integrals. These include the natural logarithm, exponential, hyperbolic, inverse trigonometric and hyperbolic functions.
• We cover several methods of integration, including u-substitution, integrals involving trigonometric functions, integration by parts, and partial fraction decomposition.

Chapter II. Fundamentals and Proofs

• We cover key ideas and skills relating to logic and mathematical thinking, proofs, formal arguments and fallacies.
• We introduce different strategies for proving results in mathematics, including proof by induction, proof by contraposition, proof by contradiction, proof by exhaustion, and proof by construction.
• We will discuss properties of sets of different number systems, including integers, real numbers, complex numbers, and vectors.
• We more rigorously define features of functions on sets, such as inverses, limits, continuity, differentiability, and integrability of functions.

Chapter III. Sequences and Series

• We introduce the concept of infinite sequences and series of real or complex numbers.
• We develop several tests for the convergence of series.
• We introduce special series such as geometric series and more general power series.
• We describe how certain functions can be represented as a series of other functions, including Taylor series, Laurent Series, and Fourier series.
• We discuss the approximation of functions using truncated forms of these series, and show how one may bound the remainder to estimate error.

Chapter IV. Linear Algebra

• We use linear algebra to understand systems of equations and subspaces, introducing rank, dimensions and bases, making these notional general through the treatment of vector spaces.
• We introduce the concepts of linear independence and linear dependence of vectors and functions and relate these to the construction of a basis.
• We solve algebraic eigenvalue problems. Eigenvalues, eigenvectors and determinants are introduced and linked with concrete applications.

Chapter V. Differential Equations

• We discuss the intuition behind, and applications of, differential equations. Initial value problems and boundary value problems are contrasted.
• We introduce methods for solving simple first order differential equations by separation of variables and the integrating factor method.
• We solve second order differential equations through the method of undetermined coefficients, reduction of order, and variation of parameters.
• We connect the solution sets of differential equation to the concept of linear independence.
• We discuss how to approximate the solutions of more complicated differential equations using series of functions.
• We solve simple eigenvalue problems for differential equations.

Teaching Arrangements

Four lectures per week.
Weekly tutorials, one hour in length.
 

Textbooks

No text books required. Learning materials will be available on the Blackboard page for this paper.

Graduate Attributes Emphasised

Scholarship, Lifelong Learning, Information Literacy, Critical Thinking, Communication
View more information about Otago's graduate attributes.

Learning Outcomes

Students who successfully complete the paper will:

• Appreciate mathematics as a modern discipline and learn what mathematicians and mathematical scientists do
• Gain increasing fluency with the processes of abstraction and generalisation in the mathematical sciences
• Begin to recognize and develop mathematical arguments and apply logic and mathematical rigor
• Manipulate mathematical expressions, derive new expressions from others and develop skills for explaining and communicating mathematical arguments

Assessment details

Final Exam: 45% of the final grade

Internal Assessment: 55% of the final grade

Internal assessment comprises assignments, tutorials, and a midterm.

Timetable