Calculus of Several Variables
University of Otago
Dunedin, New Zealand
Area of Study
Taught In English
Course Level Recommendations
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Recommended U.S. Semester Credits3 - 4
Recommended U.S. Quarter Units4 - 6
Hours & Credits
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 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 both 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 generalised to functions of more than one variable. The simplest case deals with functions of the form z=f(x,y) (ie 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 that express conservation laws, such as the conservation of energy and momentum in Physics or the conservation of mass in Chemistry.
- 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 Stokes
Demonstrate in-depth understanding of the concepts, results and methods of the paper
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