University Mathematics Handbook (2015)

IX. Vector Analysis

Chapter 11. Conservative Field in General

a.  Vector field is called conservative field if there exists scalar field such that .

b.  Vector field of class above simply connected surface body (that is, on every closed curve or there exists a two-sided surface, of which is its boundary) is conservative if, and only if, one of the following propositions hold:

1.   for every simple closed line in .

2.  Integral is independent of the line connecting points and , which is entirely in .

3.  There exists scalar field holding . In other words, the expression is a full differential. It means there exists a function such that and there holds .

4.  .

c.  If filed is conservative in and is a point in , then the potential function can be calculated using one of the following formulas: