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.
Many scientists want to predict the future state of some system, be it the state of our universe, the state of our solar system, the state of an amoeba colony, the state of our economy, the state of a chemical reaction etc. The predictions are generally based on the relationship between the rate of change of the system and properties of the system environment. In many cases the actual quantities of interest depend not only on the 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 modeling 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 modeling 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.
|Paper title||Calculus of Several Variables|
|Teaching period||Semester 2 (On campus)|
|Domestic Tuition Fees (NZD)||$955.05|
|International Tuition Fees||Tuition Fees for international students are elsewhere on this website.|
- MATH 140 or MATH 170
- MATH 251
- Schedule C
- Arts and Music, Science
- 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.
- Teaching staff
Dr Jörg Hennig
- Paper Structure
- 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
- Vector fields and line integrals
- The theorems of Green and Gauss
- Teaching Arrangements
Most weeks you will be expected to spend 3 hours in lectures, 1 hour in tutorial, and about 5 hours in assignments and self-directed activities such as reading, making summary notes, etc.
Lecture notes on the material of the course are provided on the resource page.
Highly recommended textbook:
J. Stewart, Calculus.
- Course outline
- Graduate Attributes Emphasised
- Communication, Critical thinking.
View more information about Otago's graduate attributes.
- Learning Outcomes
Students who successfully complete this paper will:
- Have the tools to deal with calculus problems in higher dimensions
- Be expected to apply them in several different contexts