Multivariable Lesson 2.1: Functions of Several Variables

So far, we have dealt with the calculus of space curves. Space curves are much like the parametric functions you dealt with in single-variable calculus, except that they also move in the -direction. For the rest of multivariable calculus, we will shift our focus to another type of function: functions of several variables. A function of a single variable takes a number as input and outputs a number. A function of two variables takes an ordered pair as input and outputs a number. This is denoted . In other words, the domain consists of ordered pairs, not the real numbers. The range is still the real numbers.

For functions of two variables, it’s convenient to represent the inputs as the - and -coordinates, and the output as the -coordinate. This way, we can visualize the function in three dimensions.

Image from the UIUC Atmospheric Sciences program

For example, imagine (or look to the right for an example of) a weather map of temperature highs which has lines (isotherms) which trace the border at which the temperature is 60, 70, 80, etc. Now imagine the coordinates on the map being the domain, and raising each point in the -direction to a height proportional to its temperature. This would be a three-dimensional graph of a function of two variables (translating the -and -coordinates into a temperature). Each isotherm would be a line traced on the graph which remains at the same -coordinate at every point, since it represents the locations in the domain at which temperature is a particular constant value.

Isotherms are a special case of what we call level curves. Level curves are curves which pass through all the locations in the domain which yield the same output. They are generated by solving for some constant . [example of graph]

We can also explore functions of three variables. These functions take three inputs (, , and ) and output a fourth. For example, while temperatures on a weather map are a function of two variables, the temperature of any exact (three-dimensional) location in a room would be a function of three variables. The domain is a three-dimensional space. Since we would need four dimensions to graph such a function, we won’t try visualizing them directly. We can, however, extend the concept of level curves to three dimensions; these so-called level surfaces represent a three-dimensional surface on which each point evaluates to the same value in the function. These level surfaces can be generated by solving for some constant . While we can’t visualize the graph itself, we can visualize its values by graphing level surfaces to get an idea of how the domain maps to the range.