Definition: a line integral \(\int\omega\) over a domain \(D\) is exact if \(\int\omega\) is dependent only on its initial and end point (independent of the path taken).
Definition: a line integral \(\int\omega\) over a domain \(D\) is closed if \(\int\omega\) it integrates to zero over every closed loop.
Theorem: Assume that the vector field (1-form) has continuous partials and the domain is simply connected (the condition can be relaxed for some directions, but we assume so for simplicity.) The following are equivalent:
- A line integral is exact.
- The line integral is closed.
- If the line integral's vector field has continuous partial derivatives, the 1-form (vector field) \(\omega\) is a gradient (i.e. has an anti-dervative)
The two non-trivial directions:
(1)=>(3): This follows because we can construct the anti-derivative (with some arbitrary starting point \(x_0\)) $$ F(\vec{x}) = \int_{\vec{x_0}}^{\vec{x}} \omega $$ The proof follows similar to the traditional fundamental theorem of calculus.
(2)=>(1): This follows from Green's theorem.
More:
One dimension
In one dimension fundamental theorem of calculus tells us that every continuous function is an exact integral.
Two dimensions In two dimensions, if a vector field is exact (i.e. is a gradient), then its mixed derivatives must be equal. i.e. if $$ \vec{F}(x,y) = \nabla U(x,y) $$ then, denoting the \(x\) and \(y\) components of \(F\) by \(P\) and \(Q\): $$ \frac{\partial P}{\partial y} = \frac{\partial Q} {\partial x} $$ This condition here is called closed.
If Green's theorem applies, then closed would imply exactness. Green's theorem in two dimensions require simple connectedness and the vector field being C1. If the domain is made small enough, then the domain can always be made simply connected. This means that closed really means locally exact. The intuition behind the simply connected requirement is exact because simply connectedness allows us to break the domain down to very small boxes where the vector field is locally exact.
When integrating a closed vector field around a very small circle, Green's theorem applies and all line integral over closed path gives zero. We call this vector field irrotational because it measures the infinitisimal tendency for the fluid to rotate if the vector field describes the fluid's velocity.
Of course there are vector fields that are not defined on simply connected regions that are still exact. The standard \(F(\vec{r}) = \frac{1}{|r|}\hat{r}\) fits this. The potential in two dimensions is \(\ln(|r|)\).
There are also vector fields that meets the closed condition who are not exact. Consider $$ \vec{F}(x,y) = \left(\frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2} \right) $$
Three dimensions
The analogue of Green's theorem becomes Stoke's theorem in 3D. Here, Stoke's theorem applies on a closed curve as long as there exists an enclosed surface with no discontinuities in the C1 vector field. So in 3D there is much more leeway to applying the "closed=>exact" logic.
Of course this all make sense because simply connectedness in 3D is an easier condition to meet in 3D than 2D. One would need a line of discontinuity, for example we could simply extend the above vector field to $$ \vec{F}(x,y,z) = \left(\frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2}, 0 \right) $$
Complex Analysis and Cauchy's Theorem
Normally when applying the two dimensional Green theorem to functions it typically requires that the partial derivatives are continuous. Cauchy's integral theorem says that holomorphic functions are exact without a priori requiring continuous partials.
