MATH203 Calculus of Several Variables
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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.
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.
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).
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
J. Stewart, Calculus (edition and version are not important)
A complete set of lecture notes is available from the resource page
Prof Jörg Frauendiener, room 223, ext 7770, firstname.lastname@example.org
3 hours per week: Tue 12:00 , Thu 12:00 and Fri 12:00
rooms to be advised
Tue 14:00, Wed 14:00, Thu 14:00
The internal assessment is made up from the ten assessed exercises.
There are ten assessed online assignments using the department's online system OLAF. The assignments count for 30% of the final mark.
The department maintains a list of possible tutors for the course. It can be obtained from the office.
Your final mark F in the paper will be calculated according to this formula:
F = 0.7E + 0.3A
- E is the Exam mark
- A is the Assignments mark
and all quantities are expressed as percentages.
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Assisting Others to Commit Academic Misconduct
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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 problemThe 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 problemIf 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.