Conservative Fields, Potential Function.

Definition 8.41   Let be a vector field defined over an open region in the 3-dimensional space. If, for any two points A and B in , the work integral is independent of the path from A to B in , we say that is a conservative field on .

Example 8.42   Take .

Definition 8.43   If is the gradient field over of some scalar function f(x,y,z), the function f is called a potential (function) for over .

Note that a potential function is not uniquely defined.

Example 8.44   Let . Then , where .

Theorem 8.45   Let be a vector field whose components M,N,P are continuous over a domain in the space. Then there exists a function f such that (i.e. a potential for ) if, and only if, is a conservative vector field.

In this case, for every two points , we have:

Theorem 8.46 (Component test)   Let be a vector field whose components M,N,P have continuous first partial derivatives over a domain in the space.

The field is conservative if, and only if

Example 8.47   Take .

1.
We have:

Hence, the field is conservative.
2.
Let's find a potential for : we look for a function f(x,y,z) such that

i.e.

We have:

where C1, C2, C3 are functions of two variables. It follows that:

3.
Take A=(0,0,0) and B=(1,2,3). We will compute the integral using two different paths:
(a)
On the segment AB:

A parametrization of the path is:

Thus:

• ;
• .

Now

(b)
On a curve: Take

Thus

Now