Review of functions of two variables: Directional derivative and the gradient

 Directional derivatives and the gradient

Review the following posts:

Directional derivatives of a function of two variables

Directional derivatives of a function of two variables (continued) 

Maximum directional derivative

The next post will be about Critical points, local extremum, absolute maxima and minima

Review of functions of two variables: Chain rule and implicit differentiation of a function of two or more variables

 In this post, we are going to review the chain rule for the partial derivatives of z = f(x,y) where x and y are functions of two independent variables u and v. Then we will review the higher order partial derivatives.

Chain rule of a function of two variables x and y where x and y are function of two independent variables

Implicit differentiation of a function of more than two variables

Next time will review Directional derivatives, maxima and minima.

Review of a function of two variables: Chain rule

 The Chain rule

• The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables.
• Three diagrams are useful for deriving formulas for functions of more than one variable, where each independent variable depends also of the other variables.

Chain rule of a function of two variables for one independent variable

Review of function 2 variables ( derivatives)

 Partial derivatives

Tangent plane and linear approximation

Review the following posts:

Derivatives of a function of two variables

Calculating partial derivatives of a function of two variables without using limit

Partial derivative value using a contour map

Tangent plane to a surface at a point

Linear approximation for function of two variables

Notion of differential of a function of two variables

Review of a function of two variables (continued)

 Objectives:

1) Review level curves and contour maps

2) Limit and continuity

Level curves and contour map

Given a function f(x,y) and a number c of its range, a level curve of the function with two variables f(x, y) for the value c is the set of points that satisfy the equation f(x, y) = c. A graph of the various level curves of a function with two variables is called a contour map.

Review the corresponding posts:

Level curves

Level curves. Making a contour map (continued)

Limit and Continuity of a function of two variables

Review the following post:

Limit of a function of two variables

Limit of a function of two variables at a boundary point

Continuity of a function of two variables

Operations on the continuity of a function of two variables

Review of functions of several variables

  Functions of several variables

I finish the chapter concerning the function of several variables and start today with a review. In this post, I review the graph of a function of 2 variables.

A. Function of 2 variables

1) Graph

a) The graph of a function of 2 variables in R³ is called a surface. It can be studied using level curves and vertical traces. A set of level curves is called a contour map.

Here are the post that refers to the graph of a function of 2 variables:

Graph of a function of 2 variables

Graph of a function of 2 variables (continued)

(to be continued)

Higher order partial derivatives of a function of two variables

 Objective: Find all second order partial derivatives of a function of two variables

High order partial derivatives

Let's consider a function of two variables. We can calculate the partial derivatives of this function. If we calculate the partial derivatives of the first partial derivatives, we obtain second order partial derivatives. If we calculate the partial derivatives of these second order partial derivatives, we obtain third order partial  derivatives and so on.

Let's consider the function:

Its partial derivatives are:

The partial derivatives of these partial derivatives are:

𝛿²f/δx² = 8x and δ²f/δy² = -8x + 30y. These new partial derivatives are called second order partial derivatives.

There are 4 second order partial derivatives of any function provided they all exist

Example

Solution

Practice

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