## 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), MATH 361 (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

### Required textbook

J. Stewart, Calculus (8th edition, metric version)

### Lecturer

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

### Lectures

3 hours per week: Tue 12:00 R7N10, Thu 12:00 R7N10 and Fri 12:00 R7N10

### Tutorials

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

### Office hours:

Tue 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

If:

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

all expressed out of 100, then your final mark F in this paper will be calculated according to this formula:

**F = 0.6E + 0.4A**

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

**Academic endeavours at the University of Otago are built upon an essential commitment to academic integrity.**

The two most common forms of academic misconduct are *plagiarism* and *unauthorised collaboration*.

#### Academic misconduct: **Plagiarism**

Plagiarism is defined as:

- Copying or paraphrasing another person’s work and presenting it as your own.
- Being party to someone else’s plagiarism by letting them copy your work or helping them to copy the work of someone else without acknowledgement.
- Using your own work in another situation, such as for the assessment of a different paper or program, without indicating the source.
- Plagiarism can be unintentional or intentional. Even if it is unintentional, it is still considered to be plagiarism.

All students have a responsibility to be aware of acceptable academic practice in relation to the use of material prepared by others and are expected to take all steps reasonably necessary to ensure no breach of acceptable academic practice occurs. You should also be aware that plagiarism is easy to detect and the University has policies in place to deal with it.

#### Academic misconduct: **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 student’s answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer.

### 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.