Functions of Several Variables

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I. Functions of two Variables and the Level Curves

 

Consider the function , what is the domain of the function ? What is the range of the function ?

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To understand the graph of , we should study the level curves of .

Definition

The level curves of a function of a two variables are the curves with equations , where is a constant.

From the plot above, we see that the level curves of are circles centered at the origin and the radius of the circle increases as the value of increases. Moreover, is not in the range of .

We can see better with the graph below :

We see that the domain of the function is {( )| } and the range of the function is [ ].

 

 

Let's take a look at the level curves of a plane .

The level curves of the plane are parallel lines. Is this the case for any other planes ?

Notice that the lighter the color of the level curve gets as the value of increases.

 

Here is the level curves of the function :

Here is the graph of the function :

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Here are the level curves of :

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Here is the graph of :

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Here are the level curves of :

 

Here is the graph of :

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Here are the level curves and graph of :

 

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Here are the level curves and graph of :

Note: Maple commands do not generate the level curves and . Why ?

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What happen to the graph of when ( ) close to ( ) ?

 

II. Functions of Three Variables and the Level Surfaces

 

Consider the function , the level surfaces of are the surfaces with equations , where is a constant.

Here are the level surfaces of :

 

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