Functions of more than two variables

 Goal: find the domain of a function of more than two variables

Function of more than two variables

Let's consider the following functions:

In the first function, each point (x,y, z) represents a point in space and the function f maps each point in space to a third quantity such as temperature or wind speed. In the second function (x,y) represents a point in the plane and t represents time. The function f maps each point in the plane to a third quantity such as pressure at a given time. 

Domain of a function of two variables

The method used for finding the domain of a function of two variables is similar to the one used for a function of one or two variables.

Examples

Solution

Practice

Absolute maxima and minima of a function of two variables

 Objective: Find the absolute maxima and minima of a function of two variables.

Absolute maxima and minima

To find the absolute maxima and minima for a function of one variable on a closed interval, we nee need the find the critical points over that interval and evaluate the function at the endpoints of the interval. In a function of two variables, the interval is replaced by a closed boundary set. A set is bounded if all of its points are found inside of a ball or a disk. In order to find the absolute extrema, we start by calculating the critical points on the boundary set and the corresponding critical values of the function f.  Then we find the maximum and minimum value of the function at the boundary set. The highest function value is the absolute maximum and the smallest function value is the absolute minimum. Before calculating these values, we need to assure that there is a possibility that they exist. The following theorem gives this assurance.

Theorem

Since we now know that a continuous function function f(x,y) defined on a closed bounded set attains its extreme values, we need to know how to find them. the following theorem allows this.

Theorem

Problem solving strategy

Finding the maximum and minimum value at the boundary set can be challenging. If the boundary set is a rectangle or a set of straight lines it is possible to parameterize the line segments and determine the maxima on each of these segments. The same approach can be used for other shapes such as circles and ellipses.

Example

Use the problem-solving strategy to find the absolute extrema of a function to determine the absolute extrema of the following function:

 

Solution

Practice

Determination of the local extremum and the saddle point of a function of two variables

 Local and global extremum in a function of two variables

The extremum of a function is either a minimum or a maximum. The purpose of determining a critical point is to determine relative maxima or minima of a function. In a function of a single variable, we determine an interval around the critical point where the value of the function for that critical point is greater or less than all the values of the function in the chosen interval. For a function of two variables we do the same consideration with the exception that the interval is a disk.

Definition 

Saddle point. Definition

Second Derivative Test. Theorem

Example

Find the critical point of the critical points of the following function. Use the second derivative test to find the local extrema.

Solution

Practice