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

1. Truth and falsehood
   We cover key ideas and skills relating to logic and mathematical thinking, proofs, formal arguments and fallacies. Concepts are developed using examples from number theory, cryptography, and propositional logic.
2. Complex numbers
   The emphasis will be on understanding the connection between Cartesian and polar representations of complex numbers, the geometric viewpoint of complex multiplication and their central importance to dynamic and physical systems.
3. Matrices and subspaces
   We explore and extend the theory and geometry of matrices and linear algebra started in MATH 130. We show how matrices are used to understand systems of equations and subspaces, introducing rank, dimensions and bases. Eigenvalues, eigenvectors and determinants are introduced and linked with concrete applications.
4. Calculus
   We extend ideas from calculus introduced in MATH 130. The toolkit is expanded significantly. Important special functions are discussed in context. We explore the Taylor approximation of a function, with key examples and applications. We reintroduce differential equations and link them with ideas from integration. We examine ways these ideas generalise to higher dimensions, revisiting partial derivatives, the gradient, and encountering some of the challenges of high dimensional integrals.

Teaching Arrangements

Four lectures per week.
Weekly tutorials, one hour in length.
Assessment is 55% internal, 45% final exam.

Textbooks

Course materials will be available for free on the resource webpage.

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 generalization 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

Timetable

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

1. Truth and falsehood
   We cover key ideas and skills relating to logic and mathematical thinking, proofs, formal arguments and fallacies. Concepts are developed using examples from number theory, cryptography, and propositional logic.
2. Complex numbers
   The emphasis will be on understanding the connection between Cartesian and polar representations of complex numbers, the geometric viewpoint of complex multiplication and their central importance to dynamic and physical systems.
3. Matrices and subspaces
   We explore and extend the theory and geometry of matrices and linear algebra started in MATH 130. We show how matrices are used to understand systems of equations and subspaces, introducing rank, dimensions and bases. Eigenvalues, eigenvectors and determinants are introduced and linked with concrete applications.
4. Calculus
   We extend ideas from calculus introduced in MATH 130. The toolkit is expanded significantly. Important special functions are discussed in context. We explore the Taylor approximation of a function, with key examples and applications. We reintroduce differential equations and link them with ideas from integration. We examine ways these ideas generalise to higher dimensions, revisiting partial derivatives, the gradient, and encountering some of the challenges of high dimensional integrals.

Teaching Arrangements

Four lectures per week.
Weekly tutorials, one hour in length.
Assessment is 55% internal, 45% final exam.

Textbooks

Course materials will be available for free on the resource webpage.

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

Timetable