## MATH203 Calculus of Several Variables

First Semester |

This paper is an introduction to the mathematics of curves, surfaces and volumes in three-dimensional space, and extends the notions of differentiation and integration to higher dimensions. It is a prerequisite for many level-300 MATH papers.

### Paper details

Many scientists spend much of their time trying to predict the future state of some system, be it the state of an oil spill, the state of our star system, the state of an amoeba colony, the state of our economy, etc. The predictions are generally based on the relationship between the rate of change of the system, or maybe the rate of change of the rate of change, and circumstances in the system environment. Usually real quantities of interest depend not only on passage of time, but on other factors as well, such as spatial variations of properties within the system and its environment. A prime example is our weather. The air pressure and the temperature change during the day and they are different in different parts of the world, so they change also in space.

Multivariate differential calculus provides the fundamental tools for modelling system changes when more than one important parameter is responsible for those changes. It is particularly fundamental to all of the physical and natural sciences, and to all situations requiring the modelling of rates of change.

In this paper, many of the ideas and techniques of one-variable differentiation and integration (as covered in MATH 160 and 170) are generalized to functions of more than one variable. The simplest case deals with functions of the form z=f(x,y), i.e., functions whose graph is a surface in three-dimensional space. Such surfaces can be drawn with the aid of level curves of the function. Paths of steepest ascent (or descent) along the surface may eventually lead to local or global extremum values of the function which generally have particular physical significance.

Other important notions covered in the paper are vector fields (such as flow fields of a fluid) and their properties and the fundamental integral identities which express conservation laws, such as the conservation of energy and momentum in Physics or the conservation of mass in Chemistry.

### Potential students

This paper is particularly relevant to Mathematics, Statistics and Physics majors, but should appeal to a wide variety of students, including those studying Computer Science, Chemistry, Surveying or any discipline requiring a quantitative analysis of systems and how they change with space and time. MATH 203 is a prerequisite for MATH 304 (Partial Differential Equations), MATH 306 (Geometry of Curves and Surfaces), COMO 303 (Numerical Analysis).

### Prerequisites

MATH 170

### Course Outline

The paper will cover the following topics:

- Vector-valued functions, vector fields, scalar fields
- Partial derivatives, directional derivatives
- Gradient, divergence and curl
- Total differential
- Taylor’s theorem for functions of several variables
- Inverse and implicit function theorems
- Local extrema, Lagrange multipliers
- Integrals over regions in two and three dimensions
- Mean value theorems for functions of several variables
- Iterated integrals
- Change of variables
- The theorems of Green and Gauss

### Recommended textbook

J. Stewart, Calculus (edition and version are not important)

### Lecture notes

A complete set of lecture notes is available from the resource page

### Lecturer

Prof Jörg Frauendiener, room 223, ext 7770, joergf@maths.otago.ac.nz

### Lectures

3 hours per week: Tue 12:00 Arch3, Thu 12:00 and Fri 12:00 Moot Court Richardson Building

### Tutorials

Tue 14:00 MA242, Wed 14:00 MA242, Thu 14:00 MA242

### Office hours:

Wed 10-12

### Internal Assessment

The internal assessment is made up from the ten assessed exercises.

### Assignments

There are ten assessed online assignments using the department's online system OLAF. The assignments count for 30% of the final mark.

### Tutors

The department maintains a list of possible tutors for the course. It can be obtained from the office.

### Final mark

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

**F = 0.7E + 0.3A**

where:

- E is the Exam mark
- A is the Assignments 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.

### Sample problem

Neglecting inertial forces, a snow avalanche roughly follows a path of steepest descent down a mountainside. Calculate the path of steepest descent when the terrain is represented by an altitude function of the form h(x,y).

### Sample problem

The container for a commercial beverage is to be manufactured with a specified volume. Find the dimensions of a cylindrical container which minimizes the amount of aluminium required for each can.### Sample problem

If R is the total resistance of three resistors connected in parallel, find the maximum error in the value of R if the measured values of the three resistors have possible errors of 0.5%.### Geoffery Taylor...

..., 1886-1975, was the grandson of George Boole. He was one of the most distinguished physical scientists of last century, using his deep insight and originality and mathematical skill to increase greatly our understanding of phenomena such as the turbulent flow of fluids.His interest in the science of fluid flow was not confined to theory; he was one of the early pioneers of aeronautics, and designed a new type of anchor, now widely used in small boats throughout the world, that came about through his passion for sailing. Taylor spent most of his working life in the Cavendish Laboratory in Cambridge, where he investigated the mechanics of fluid and solid materials; his discoveries and ideas have had application throughout mechanical, civil and chemical engineering, meteorology, oceanography and material science. He was also a noted research leader, and his group in Cambridge became one of the most productive centres for the study of fluid mechanics.