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Tuesday, 26 March 2013

line integrals part 5


we saw that if we knew that the vector field  was conservative then  was independent of path.  This in turn means that we can easily evaluate this line integral provided we can find a potential function for .

In this section we want to look at two questions.  First, given a vector field  is there any way of determining if it is a conservative vector field?  Secondly, if we know that  is a conservative vector field how do we go about finding a potential function for the vector field?

The first question is easy to answer at this point if we have a two-dimensional vector field.  For higher dimensional vector fields we’ll need to wait until the final section in this chapter to answer this question.  With that being said let’s see how we do it for two-dimensional vector fields.

Theorem
Let  be a vector field on an open and simply-connected region D.  Then if P and Q have continuous first order partial derivatives in D and
                                                                 
the vector field  is conservative.

Let’s take a look at a couple of examples.

Example 1  Determine if the following vector fields are conservative or not.
(a)    [Solution]
(b)    [Solution]

Solution
Okay, there really isn’t too much to these.  All we do is identify P and Q then take a couple of derivatives and compare the results.

(a) 

In this case here is P and Q and the appropriate partial derivatives.
                                               
So, since the two partial derivatives are not the same this vector field is NOT conservative.
(b) 

Here is P and Q as well as the appropriate derivatives.
            

The two partial derivatives are equal and so this is a conservative vector field.

Now that we know how to identify if a two-dimensional vector field is conservative we need to address how to find a potential function for the vector field.  This is actually a fairly simple process.  First, let’s assume that the vector field is conservative and so we know that a potential function,  exists.  We can then say that,



Or by setting components equal we have,


By integrating each of these with respect to the appropriate variable we can arrive at the following two equations.


We saw this kind of integral briefly at the end of the section on iterated integrals in the previous chapter.

It is usually best to see how we use these two facts to find a potential function in an example or two.

Example 2  Determine if the following vector fields are conservative and find a potential function for the vector field if it is conservative.
(a)    [Solution]
(b)    [Solution]

Solution
(a) 
Let’s first identify P and Q and then check that the vector field is conservative..
                                             

So, the vector field is conservative.  Now let’s find the potential function.  From the first fact above we know that,
                                          

From these we can see that
                     
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