Calculus III:

Regions and paths : definitions
Properties

1. Definitions

Given a continuous vector field in some domain D.

• is a conservative vector field if there is a function f such that = ∇f. The function f is called a potential function for the vector field.

• ∫_C . is independent of the path if ∫_C1 . = ∫_C2 . for any two paths C1 and C2 in the region D with the same initial and final points.

• A path C is simple if it doesn’t cross itself. A circle is a simple path.

• A path C is closed if its initial and final points are the same point. A circle is a closed path.

• A region D is open if it doesn’t contain any of its boundary points.

• A region D is connected if we can connect any two points in the region with a path that lies completely in D.

• A region D is simply-connected if it is connected and it contains no holes.

2. Properties

• ∫_C ∇f . is independent of path.

• If is a conservative vector field then is independent of path.

• If is a continuous vector field on an open connected region D and if ∫_C. is independent of path (for any path in D) then is a conservative vector field on D.

• If ∫_C. is independent of path then ∫_C . = 0 for every closed path C.

• If ∫_C. = 0 for every closed path C then ∫_C . is independent of path.