Level surface of a function of three variables

Objective:  Find the level surface of a function of three variables.

In a function of two variables, we have level curves which are defined in the xy plane. In a function of three variables, level curves are called surfaces

Definition

Level surfaces in a function of three variables

There are several types of level surfaces depending on the form of the function of three variables. Let's limit ourselves to three types.: planes, quadric surfaces and spheres.  

Plane

When the function f(x,y,z) is a linear function leading to the equation  ax + by + cz = d where a,b, c and are contants, the level surface is a plane. Planes are flat, have no curvature and extend infinitely.

Quadric surfaces

If the function f(x,y,z) is a quadric function leading to the equation ax² + by² + cz² = d which is a second degree polynomial, the level surface is a quadric surface. Quadric surfaces can be ellipsoid, hyperboloid, paraboloid, cone and cylinder depending on the form of the quadric equation.

Ellipsoid

Hyperboloid

There are two types:

1.Hyperboloid of one sheet

 

2. Hyperboloid of 2 sheets

Consists of two separate surfaces like two bowls facing each other.

Equation:

Paraboloid

There are two types:

 

Cone

Sphere

Example

Solution

Practice

Find the level surface of the function of 3 variables and describe it:

a) 

 

b) 

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

Critical points in a function of two variables

 Objective: Use partial derivatives to locate critical points for a function of two variables

Critical points. Definition

Examples

Solution

Next, we set each of these expressions equal to zero:

Therefore x = 2 and y = -3, (2, -3) is a critical point of f.

We must also check for the possibility that the denominator of each partial derivative can be equal to zero. In this case, the partial derivative doesn't exist. Since the denominator is the same in both partial derivatives, we need to do this once.

This equation represents also an hyperbola. We should also note that the domain of consists of points satisfying the inequality:

Therefore, any points on the hyperbola are not only critical points, they are also at the boundary of the domain. Let's put the equation of the hyperbola in standard form by completing the square:

Notice that (2, -3) is the center of the hyperbola.

We make each partial derivative equal to zero which give a system of equations with x and y. 

Subtracting the second equation from the first gives: 10 y + 10 = 0  y= -1. Substiuting y in the first

equation gives 2x -2 + 4 = 0, 2x + 2 = 0, x = -1. Therefore (-1, -1) is a critical point of the given function. There are no points in R² that make either partial derivative not to exist since both of them are defined for any point (x, y).

Practice