Partial derivatives of a function of three variables

 The partial derivative of a function of three variables can be calculated the same way we calculate the partial derivatives of a function of two variables. To calculate the partial derivative with respect to x, we consider y and z constants then we calculate the derivative considering only x as a variable. In general to calculate the partial derivative to one variable, we calculate the derivative with respect the considered variable while we keep the other two variables constant.

Definition

Example

Solution

To  calculate the partial derivative with respect to x using the limit definition, let's start by using the formula                                                                                                                                              

Let's start by calculating f(x + h, y, z):                                                                                                         

Let's substitute f(x+h, y, z) and f(x,y,z) in the formula:                                                                                   

                                                                                                     

Practice                                                                                                                                                     

Limit of a function of three or more variables

 Function of three or more variables

Let's consider two functions respectively of three and four variables that have applications in the real world. Let' f(x,y,z) be a function that measures the temperature at a location (x,y,z) and f(x,y,z,t) a function that measures the air pressure at a time t. How can we take the limit of a function in R³? What does it mean for a function of three or more variables to be continuous?

The answers to these questions start by extending the notion of a δ disk in a function of two variables to functions of three or more variables. The notions of limits and continuity for these functions are similar to those of a function of two variables.

Definition of a δ ball

The notion of δ intervenes in the definition of a function in any dimension. It is a neighborhood located in the domain of definition of a function. In a function of one variable, it is represented by a line. This line represents al the points x of which the distance from a central point is less than ẟ. In a function of two variables, it is represented by a disk. This disk represents all the points located within the distance of the radius of a circle centered a point (x₀, y₀) than In a function of two or more variables it is represented by a ball which represents all the points located with the distance of the radius of the ball. The ball is also called sphere. Here is the definition:

 

Limit of a function of three or more variables

The limit of a function implies two neighborhoods: one in the domain of the function and the other in the range. To show that the limit of a function at a point located in the domain of the function exists, we need to find a neighborhood located around that point such that for any point located in that neighborhood the value of the function at that point is close to a fixed value  called the limit of the function. This consideration is valuable for a function in any dimension.

In two dimensions the definition states as follows:

In three dimensions we have:

In 4 dimensions we have:

Example 

Solution

Practice

Find the limit of the following function:

 

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