Review of functions of more than 2 variables

In the previous post we reviewed the function of two variables. In this post we will review certain notions regarding the functions of more than 2 variables:

Functions of more than two variables

Level surface of a function of three variables

Limit of a function of three or more variables

Partial derivatives of a function of three variables

Review of a function of two variables: Maxima and Minima

 Maxima and Minima Problems

In this post we review the following notions:

Critical points in a function of three variables

Local extremum and the saddle point in a function of two variables

Absolute maxima and minima

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)