Definitions
First suppose that is a continuous vector field in some domain D.
1. is a conservative vector field if there is a function f such that . The function f is called a potential function for the vector field. We first saw this definition in the first section of this chapter.
2. is independent of path if for any two paths and in D with the same initial and final points.
3. A path C is called closed if its initial and final points are the same point. For example a circle is a closed path.
4. A path C is simple if it doesn’t cross itself. A circle is a simple curve while a figure 8 type curve is not simple.
5. A region D is open if it doesn’t contain any of its boundary points.
6. A region D is connected if we can connect any two points in the region with a path that lies completely in D.
7. A region D is simply­connected if it is connected and it contains no holes. We won’t need this one until the next section, but it fits in with all the other definitions given here so this was a natural place to put the definition.
Facts
1. is independent of path.
This is easy enough to prove since all we need to do is look at the theorem above. The theorem tells us that in order to evaluate this integral all we need are the initial and final points of the curve. This in turn tells us that the line integral must be independent of path.
2. If is a conservative vector field then is independent of path.
This fact is also easy enough to prove. If is conservative then it has a potential function, f, and so the line integral becomes . Then using the first fact we know that this line integral must be independent of path.
3. If is a continuous vector field on an open connected region D and if is independent of path (for any path in D) then is a conservative vector field on D.
4. If is independent of path then for every closed path C.
5. If for every closed path C then is independent of path.
These are some nice facts to remember as we work with line integrals over vector fields. Also notice that 2 & 3 and 4 & 5 are converses of each other.
Title: May 7­11:10 AM (1 of 9)
The fundamental theorem of Calculus
Title: May 12­11:33 AM (2 of 9)
so...this says we can evaluated the line integral of a conservative vector field
Title: May 12­12:38 PM (3 of 9)
line integral of conservative vector field depends only on initial and terminal points of a curve
so line integrals of conservative vector fields are independent of path
Title: May 12­12:50 PM (4 of 9)
Title: May 12­12:57 PM (5 of 9)
Two more theorems
Title: May 12­12:59 PM (6 of 9)
Title: May 12­1:08 PM (7 of 9)
Title: May 12­1:12 PM (8 of 9)
Title: May 13­2:09 PM (9 of 9)