Chapter 15
Section 15.3: Conservative Vector Fields and Independence of Path

• Informal: The work done by a conservative vector field from one point to another is the same for all paths between those points.
• Formal: Fundamental Theorem of Line Integrals on a conservative vector field:
  Let `C` be a piecewise smooth curve lying in an open region `R` and given by `bbr(t)=x(t)bbi + y(t)bbj +z(t)bbk, a <= t <= b`.
  If `bbF(x,y) = Mbbi + Nbbj` is conservative in `R`, and `M` and `N` are continuous in `R`, then
  `int_CbbF•dbbr = int_C grad f•dbbr = f(x(b),y(b)) - f(x(a),y(a)) ` where `f` is a potential function of `bbF`, i.e., `bbF(x,y) = grad f(x,y).`
• General: Independence of Path and Conservative Vector Fields:
  If `bbF` is continuous on an open connected region, then the line integral
  `int_C bbF • dbbr`
  is independent of path if and only if `bbF` is conservative.
• Equivalent Conditions
  Let `bbf(x,y,z) = Mbbi +Nbbj +Pbbk` have continuous first partial derivatives in an open connected region `R`, and let `C` be a piecewise smooth curve in `R`. The following conditions are equivalent.
  1. `bbF` is conservative, i.e., `bbF = grad f` for some function `f`,
  2. `int_c bbF • dbbr ` is independent of path.
  3. `int_C bbF • dbbr = 0` for every closed curve `C` in `R`.