## MATH140 Fundamentals of Modern Mathematics 2

Second Semester |

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.

### Paper details

MATH 140 focuses on both ideas and methods. The material will be broken into several sections:

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.

### Potential students

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.

### Learning Outcomes

- 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.

### Prerequisites

MATH 130

### Main topics

**Truth and Falsehood**

- Sets and functions
- Set notation, the empty set, union and intersection, subsets, complement, exclusion, sets of natural numbers, integers, rational numbers, real numbers
- De Morgan's Laws
- Logic, Propositional logic, Quantifiers
- Introduction to Proof
- What is a proof, basic proof strategies, direct proof, contradiction, contrapositive, case by case
- Induction

**Complex Variables**

- Complex numbers
- Motivation (complex roots), conjugate, real part and imaginary part, complex plane, sums and products of complex numbers
- Fundamental theorem of algebra
- Euler's identity, polar representation
- Applications
- Complex numbers as vectors
- The complex plane

**Matrices and Subspaces**

- Vectors in 2, 3 and n dimensions, addition/subtraction, scalar product, lines and planes, matrix definition, transpose, addition/subtraction, scalar multiplication, matrix multiplication
- Cross product, Scalar and vector triple product
- Linear independence
- Subspaces, basis, dimension, rank
- Algebraic properties of matrix addition, scalar multiplication, matrix multiplication, and the transpose
- Diagonal, triangular and symmetric matrices
- Inverse matrices, uniqueness of inverse, properties of inverse, negative matrix powers
- Elementary matrices
- Fundamental theorem of invertible matrices
- Gauss-Jordan method for computing the inverse
- Determinants
- Linear transformations and eigenvectors

**Calculus**

- Special functions: Natural logarithm, exponential, hyperbolic, inverse trigonometric and hyperbolic functions
- Methods of integration: u-substitution, integrals involving trigonometric functions, integration by parts, integration of rational functions, partial fraction decomposition
- Infinite sequences and series, series convergence tests, geometric series
- Power series, Taylor series, approximation of functions
- Differential equations: classification, first-order separable, first-order linear, second-order constant coefficient, iterative and numerical methods for solving differential equations
- Functions of two and three variables, limits and rates of change
- Partial derivatives, directional derivatives, the Laplacian operator, applications

### Texts

Course materials will be available on the resource page. These will be posted as the course progresses.

### Lecturers and Contact

General inquiries: Math140@maths.otago.ac.nz

First six weeks of lectures: Dr Timothy Candy (Office: Science III, room 216. email: tcandy@maths.otago.ac.nz)

Second six weeks of lectures and paper coordinator: Dr R. A. Van Gorder (Office: Science III, room 212. email: rvangorder@maths.otago.ac.nz)

### Lectures

Monday, Tuesday, Wednesday, Thursday, at 12 noon

### Tutorials

There will be 10 tutorials, scheduled at different times during the week. Attendance is mandatory, and there will be a tutorial activity to complete based on what is covered in prior lectures.

### Internal Assessment

The total of all internal assessment makes up 55% of your mark, and is comprised of the following:

- Written homework assignments (A) are worth 20% of your final mark.
- Online quizzes (Q) make up 10% of your final mark.
- Tutorial activities (T) make up 10% of your final mark.
- A written midterm exam (M) will make up 15% of your mark.

The remaining 45% of your mark comes from the final exam (E).

You can check your marks by clicking on the Resources link at the top of this page.

### Terms Requirement

None.

### Exam format

The 3-hour final exam is answered in spaces provided on the question booklet. All questions should be attempted and the number of marks available for each question is indicated on the paper.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = 0.45*E + 0.2*A + 0.1*Q + 0.1*T + 0.15*M**

where:

- E is the Exam mark
- A is the Assignments mark
- Q is the Quizzes mark
- T is the Tutorial assignments mark
- M is the Midterm exam mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.